Permutation and Combination

Permutation and combination are two fundamental concepts in mathematics that deal with the arrangement and selection of elements from a set.

Permutation refers to the arrangement of elements in a specific order, while combination refers to the selection of elements without regard to order.

In permutation, the number of possible arrangements is determined by the formula nPr = n! / (n - r)!, where n is the total number of elements and r is the number of elements to be arranged.

In combination, the number of possible selections is determined by the formula nCr = n! / (r! * (n - r)!).

Permutation is used in situations where the order of elements matters, such as in determining the number of ways to arrange letters in a word or the number of ways to select a committee from a group of people.

Combination is used in situations where the order of elements does not matter, such as in determining the number of ways to select a group of friends from a class or the number of ways to choose a lottery ticket.

What is Permutation?

Permutation is a mathematical operation that involves arranging a set of objects in a specific order. The number of possible permutations of a set of n objects is given by the formula n!, where n! represents the factorial of n.

For example, if we have a set of three objects, A, B, and C, the possible permutations of these objects are:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

As you can see, there are 6 possible permutations of 3 objects.

Permutations are used in a variety of applications, including:

• Counting the number of possible outcomes in a given situation
• Generating random samples
• Designing experiments
• Solving puzzles

Here are some additional examples of permutations:

• The number of ways to choose a president, vice president, and secretary from a group of 10 people is 10P3 = 720.
• The number of ways to arrange the letters in the word “APPLE” is 5! = 120.
• The number of ways to choose 5 cards from a standard deck of 52 cards is 52C5 = 2,598,960.

Permutations are a fundamental concept in mathematics and have a wide range of applications in the real world.

What is a Combination?

Combinations

In mathematics, a combination is a selection of items from a set where the order of the items does not matter. For example, if you have a set of three letters, {A, B, C}, then the following are all combinations of two letters from the set:

• AB
• AC
• BC

The number of combinations of n items taken r at a time is given by the formula:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

where:

• n is the total number of items in the set
• r is the number of items to be selected

For example, the number of combinations of 5 items taken 3 at a time is:

$$C(5, 3) = \frac{5!}{3!2!} = \frac{120}{6 \cdot 2} = 10$$

This means that there are 10 different ways to select 3 items from a set of 5 items.

Examples of Combinations

Combinations are used in a variety of applications, including:

• Counting: Combinations can be used to count the number of possible outcomes in a given situation. For example, if you have a deck of 52 cards, the number of possible 5-card hands is given by:

$$C(52, 5) = \frac{52!}{5!47!} = 2,598,960$$

This means that there are 2,598,960 different ways to select a 5-card hand from a deck of 52 cards.

• Probability: Combinations can be used to calculate the probability of an event occurring. For example, if you roll a six-sided die, the probability of rolling a 6 is 1/6. This is because there are 6 possible outcomes, and only one of them is a 6.

• Statistics: Combinations are used in a variety of statistical applications, such as hypothesis testing and confidence intervals.

Conclusion

Combinations are a powerful tool that can be used to solve a variety of problems. They are used in a variety of applications, including counting, probability, and statistics.

Permutation and Combination Formulas

Permutation and Combination Formulas

In mathematics, permutations and combinations are two fundamental concepts used to determine the number of possible arrangements or selections from a given set of elements. These formulas are widely applied in various fields, including probability, statistics, computer science, and many real-world scenarios.

Permutation Formula

The permutation formula calculates the number of possible arrangements of a given number of elements taken all at once. It is denoted as P(n, r) or nPr, where:

• n represents the total number of elements in the set.
• r represents the number of elements to be selected from the set.

The permutation formula is given by:

P(n, r) = n! / (n - r)!

where “!” denotes the factorial function. The factorial of a non-negative integer n is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Example:

Suppose you have a group of 5 friends and you want to select 3 of them to form a committee. The number of possible committees can be calculated using the permutation formula:

P(5, 3) = 5! / (5 - 3)! = 5! / 2! = (5 × 4 × 3 × 2 × 1) / (2 × 1) = 60

Therefore, there are 60 possible ways to select a committee of 3 friends from the group of 5.

Combination Formula

The combination formula calculates the number of possible selections of a given number of elements from a set, without regard to the order of selection. It is denoted as C(n, r) or nCr, where:

• n represents the total number of elements in the set.
• r represents the number of elements to be selected from the set.

The combination formula is given by:

C(n, r) = n! / (r! × (n - r)!)

Example:

Continuing with the previous example, suppose you want to select 3 friends from the group of 5 to invite to a dinner party. The number of possible invitations can be calculated using the combination formula:

C(5, 3) = 5! / (3! × (5 - 3)!) = 5! / (3! × 2!) = (5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1) × (2 × 1)) = 10

Therefore, there are 10 possible ways to select 3 friends from the group of 5 for the dinner party.

Key Differences:

• Permutations consider the order of selection, while combinations do not.
• The permutation formula involves dividing by (n - r)!, while the combination formula involves dividing by both r! and (n - r)!.
• Permutations are used when the order matters, such as in arranging objects in a specific sequence. Combinations are used when the order does not matter, such as in selecting a group of items without regard to their order.

In summary, permutation and combination formulas provide systematic methods for determining the number of possible arrangements or selections from a given set of elements. These formulas have extensive applications in various fields and practical scenarios.

Difference Between Permutation and Combination

Permutation and combination are two fundamental concepts in mathematics, particularly in the field of combinatorics. Both involve selecting items from a set, but they differ in the way the selections are made and the order in which the items are arranged.

Permutation

In a permutation, the order of the items matters. Each arrangement of the items is considered a distinct permutation. For example, if we have the letters A, B, and C, the following are all permutations of these letters:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

In general, the number of permutations of n distinct items is given by the formula:

$$P(n) = n!$$

where n! (read as “n factorial”) is the product of all positive integers up to n. For example, the number of permutations of 3 items is:

$$P(3) = 3! = 3 \times 2 \times 1 = 6$$

Combination

In a combination, the order of the items does not matter. Each set of items is considered a distinct combination. For example, if we have the letters A, B, and C, the following are all combinations of these letters, taken 2 at a time:

• AB
• AC
• BC

In general, the number of combinations of n items, taken r at a time, is given by the formula:

$$C(n, r) = \frac{P(n)}{r!(n-r)!}$$

where P(n) is the number of permutations of n items, r! is the factorial of r, and (n-r)! is the factorial of n-r. For example, the number of combinations of 3 items, taken 2 at a time, is:

$$C(3, 2) = \frac{P(3)}{2!(3-2)!} = \frac{6}{2!1!} = 3$$

Summary

The following table summarizes the key differences between permutation and combination:

Applications

Permutation and combination have numerous applications in various fields, including:

• Probability: Permutation and combination are used to calculate probabilities of events involving the selection of items from a set.
• Statistics: Permutation and combination are used in statistical sampling and hypothesis testing.
• Computer science: Permutation and combination are used in algorithms for generating permutations and combinations, as well as in cryptography and coding theory.
• Mathematics: Permutation and combination are used in various branches of mathematics, such as algebra, number theory, and graph theory.

Understanding the concepts of permutation and combination is essential for solving a wide range of problems in mathematics and its applications.

Uses of Permutation and Combination

Permutations and combinations are two fundamental concepts in mathematics that deal with the arrangement and selection of elements from a set. They find numerous applications in various fields, including probability, statistics, computer science, and genetics. Let’s explore their uses with examples:

Permutations:

1. Counting Arrangements: Permutations are used to determine the number of possible arrangements of elements in a specific order. For instance, if you have three letters A, B, and C, the number of permutations of these letters taken 2 at a time is 3P2 = 3! / (3-2)! = 6. This means there are six possible arrangements: AB, AC, BA, BC, CA, and CB.

2. Ordering Objects: Permutations are crucial in situations where the order of objects matters. For example, in a race with 10 runners, the number of possible permutations for the top three positions is 10P3 = 10! / (10-3)! = 720. This represents the different ways the top three runners can be arranged.

3. Password Combinations: Permutations are used to generate secure passwords. By considering the possible permutations of characters, including letters, numbers, and symbols, a vast number of unique passwords can be created, making it harder for unauthorized access.

Combinations:

1. Selecting Items: Combinations are used to determine the number of ways to select a specific number of items from a set without regard to their order. Continuing with the example of letters A, B, and C, the number of combinations of these letters taken 2 at a time is 3C2 = 3! / (2! * (3-2)!) = 3. This means there are three possible combinations: AB, AC, and BC.

2. Selecting Committees: Combinations are useful in forming committees or groups from a larger population. For instance, if a club with 20 members needs to select a committee of 5, the number of possible combinations is 20C5 = 20! / (5! * (20-5)!) = 15,504.

3. Probability Calculations: Combinations play a vital role in probability calculations. They help determine the likelihood of specific events occurring in random selections. For example, in a lottery with 100 tickets, if 10 tickets are drawn, the probability of winning can be calculated using combinations.

4. Genetics: In genetics, combinations are used to study the inheritance of traits. They help determine the possible combinations of alleles that can be passed from parents to offspring, influencing genetic diversity and the likelihood of specific traits appearing.

These are just a few examples of the many uses of permutations and combinations. Their applications extend to various fields, providing a powerful tool for analyzing and solving problems involving arrangements and selections.

Solved Examples of Permutation and Combinations

Solved Examples of Permutation and Combinations

Permutation

Example 1: How many different ways can you arrange the letters in the word “apple”?

Solution: There are 5 letters in the word “apple”, so we can use the permutation formula:

$$P(n, r) = \frac{n!}{(n-r)!}$$

where:

• n is the total number of items
• r is the number of items to be selected

In this case, n = 5 and r = 5, so:

$$P(5, 5) = \frac{5!}{(5-5)!} = \frac{5!}{0!} = 120$$

Therefore, there are 120 different ways to arrange the letters in the word “apple”.

Example 2: A club has 10 members. How many different ways can they elect a president, vice president, and secretary?

Solution: In this case, we need to select 3 people from a group of 10. So, we can use the permutation formula:

$$P(n, r) = \frac{n!}{(n-r)!}$$

where:

• n is the total number of items
• r is the number of items to be selected

In this case, n = 10 and r = 3, so:

$$P(10, 3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} = 720$$

Therefore, there are 720 different ways to elect a president, vice president, and secretary from a club of 10 members.

Combination

Example 1: A box contains 10 red balls and 5 blue balls. How many different ways can you select 3 balls from the box if you must select at least 1 red ball?

Solution: In this case, we need to select at least 1 red ball from a group of 10 red balls and 2 other balls from a group of 5 blue balls. So, we can use the combination formula:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

where:

• n is the total number of items
• r is the number of items to be selected

In this case, n = 10 + 5 = 15 and r = 3, so:

$$C(15, 3) = \frac{15!}{3!(15-3)!} = \frac{15!}{3!12!} = 455$$

Therefore, there are 455 different ways to select 3 balls from the box if you must select at least 1 red ball.

Example 2: A company has 10 employees. How many different ways can they select a team of 5 employees to work on a project?

Solution: In this case, we need to select 5 employees from a group of 10 employees. So, we can use the combination formula:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

where:

• n is the total number of items
• r is the number of items to be selected

In this case, n = 10 and r = 5, so:

$$C(10, 5) = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = 252$$

Therefore, there are 252 different ways to select a team of 5 employees from a company of 10 employees.

Permutation and Combination – Practice Questions

Permutation and Combination – Practice Questions

1. A bag contains 10 red balls, 15 blue balls, and 20 green balls. How many ways can you select 5 balls from the bag if:

a) There are no restrictions?

b) You must select at least 1 red ball?

c) You must select exactly 2 blue balls?

Solution:

a) To calculate the total number of ways to select 5 balls from the bag without any restrictions, we can use the combination formula:

C(n, r) = n! / (n - r)! / r!

where n is the total number of items, r is the number of items to be selected, and ! denotes the factorial function.

In this case, n = 10 + 15 + 20 = 45 and r = 5.

C(45, 5) = 45! / (45 - 5)! / 5! = 45! / 40! / 5! = 1,221,759

Therefore, there are 1,221,759 ways to select 5 balls from the bag without any restrictions.

b) To calculate the number of ways to select 5 balls from the bag with at least 1 red ball, we can first select 1 red ball from the 10 red balls. This can be done in 10 ways.

Once we have selected 1 red ball, we need to select 4 more balls from the remaining 34 balls (15 blue balls + 20 green balls). This can be done in C(34, 4) ways.

Therefore, the total number of ways to select 5 balls from the bag with at least 1 red ball is:

10 * C(34, 4) = 10 * 10,626 = 106,260

c) To calculate the number of ways to select 5 balls from the bag with exactly 2 blue balls, we can first select 2 blue balls from the 15 blue balls. This can be done in C(15, 2) ways.

Once we have selected 2 blue balls, we need to select 3 more balls from the remaining 30 balls (10 red balls + 20 green balls). This can be done in C(30, 3) ways.

Therefore, the total number of ways to select 5 balls from the bag with exactly 2 blue balls is:

C(15, 2) * C(30, 3) = 105 * 4060 = 426,300

2. A committee of 5 people is to be formed from a group of 10 men and 8 women. How many different committees are possible if:

a) There are no restrictions?

b) The committee must include at least 2 women?

c) The committee must have an equal number of men and women?

Solution:

a) To calculate the total number of ways to form a committee of 5 people from a group of 10 men and 8 women without any restrictions, we can use the combination formula:

C(n, r) = n! / (n - r)! / r!

In this case, n = 10 + 8 = 18 and r = 5.

C(18, 5) = 18! / (18 - 5)! / 5! = 18! / 13! / 5! = 8,568

Therefore, there are 8,568 different committees that can be formed from the group of 10 men and 8 women without any restrictions.

b) To calculate the number of ways to form a committee of 5 people from the group with at least 2 women, we can first select 2 women from the 8 women. This can be done in C(8, 2) ways.

Once we have selected 2 women, we need to select 3 more people from the remaining 16 people (10 men + 6 women). This can be done in C(16, 3) ways.

Therefore, the total number of ways to form a committee of 5 people from the group with at least 2 women is:

C(8, 2) * C(16, 3) = 28 * 560 = 15,680

c) To calculate the number of ways to form a committee of 5 people from the group with an equal number of men and women, we can first select 3 men from the 10 men. This can be done in C(10, 3) ways.

Once we have selected 3 men, we need to select 2 women from the 8 women. This can be done in C(8, 2) ways.

Therefore, the total number of ways to form a committee of 5 people from the group with an equal number of men and women is:

C(10, 3) * C(8, 2) = 120 * 28 = 3,360

Frequently Asked Questions on Permutations and Combinations

What do you mean by permutations and combinations?

Permutations and combinations are two fundamental concepts in mathematics that deal with the arrangement and selection of elements from a set. While both involve selecting items from a set, they differ in the way the selection is made and the order in which the elements are considered.

Permutations

In a permutation, the order in which elements are selected and arranged matters. Each distinct arrangement of elements is considered a different permutation. The number of permutations of n distinct objects taken r at a time is given by the formula:

$$P(n, r) = n! / (n - r)!$$

where:

• P(n, r) represents the number of permutations of n objects taken r at a time.
• n! represents the factorial of n, which is the product of all positive integers up to n.
• (n - r)! represents the factorial of n - r, which is the product of all positive integers from n - r down to 1.

Example:

Suppose you have a set of 4 distinct letters: A, B, C, and D. To find the number of permutations of these letters taken 2 at a time, we use the formula:

$$P(4, 2) = 4! / (4 - 2)!$$

$$P(4, 2) = 4! / 2!$$

$$P(4, 2) = (4 \times 3 \times 2 \times 1) / (2 \times 1)$$

$$P(4, 2) = 12$$

Therefore, there are 12 different permutations of 4 letters taken 2 at a time. These permutations include:

• AB
• AC
• AD
• BA
• BC
• BD
• CA
• CB
• CD
• DA
• DB
• DC

Combinations

In a combination, the order in which elements are selected does not matter. Only the selection of elements is considered. The number of combinations of n distinct objects taken r at a time is given by the formula:

$$C(n, r) = n! / (r! \times (n - r)!)$$

where:

• C(n, r) represents the number of combinations of n objects taken r at a time.
• n! represents the factorial of n, which is the product of all positive integers up to n.
• r! represents the factorial of r, which is the product of all positive integers up to r.
• (n - r)! represents the factorial of n - r, which is the product of all positive integers from n - r down to 1.

Example:

Using the same set of 4 distinct letters: A, B, C, and D, let’s find the number of combinations of these letters taken 2 at a time. We use the formula:

$$C(4, 2) = 4! / (2! \times (4 - 2)!)$$

$$C(4, 2) = 4! / (2! \times 2!)$$

$$C(4, 2) = (4 \times 3 \times 2 \times 1) / ((2 \times 1) \times (2 \times 1))$$

$$C(4, 2) = 6$$

Therefore, there are 6 different combinations of 4 letters taken 2 at a time. These combinations include:

• AB
• AC
• AD
• BC
• BD
• CD

In summary, permutations consider the order of selection and arrangement, while combinations only consider the selection of elements without regard to order. Permutations are used when the order matters, such as in arranging letters to form words or determining the sequence of events. Combinations are used when the order does not matter, such as selecting a group of people from a larger group or choosing a set of items from a collection.

Give examples of permutations and combinations.

Permutations

In mathematics, a permutation is an arrangement of elements of a set in a specific order. For example, if we have the set {1, 2, 3}, the permutations of this set are:

• (1, 2, 3)
• (1, 3, 2)
• (2, 1, 3)
• (2, 3, 1)
• (3, 1, 2)
• (3, 2, 1)

The number of permutations of a set with n elements is given by the formula n!. So, in the example above, there are 3! = 6 permutations of the set {1, 2, 3}.

Combinations

In mathematics, a combination is a selection of elements from a set without regard to order. For example, if we have the set {1, 2, 3}, the combinations of this set are:

• (1, 2)
• (1, 3)
• (2, 3)

The number of combinations of a set with n elements taken r at a time is given by the formula nCr = n! / (r!(n - r)!). So, in the example above, there are 3C2 = 3! / (2!1!) = 3 combinations of the set {1, 2, 3} taken 2 at a time.

Examples of Permutations and Combinations

Here are some examples of permutations and combinations in real life:

• Permutations:
  • The order in which people finish a race.
  • The order in which letters are arranged in a word.
  • The order in which cards are dealt from a deck.
• Combinations:
  • The different ways to choose 5 people from a group of 10.
  • The different ways to choose 3 flavors of ice cream from a list of 10.
  • The different ways to choose 2 toppings for a pizza from a list of 5.

Permutations and combinations are used in a variety of applications, including probability, statistics, and computer science.

What is the formula for permutations and combinations?

Permutations

In mathematics, a permutation is an arrangement of a set of objects in a specific order. The number of permutations of a set of n objects is given by the formula:

$$P(n, r) = \frac{n!}{(n-r)!}$$

where:

• P(n, r) is the number of permutations of n objects taken r at a time
• n! is the factorial of n, which is the product of all positive integers up to n
• (n-r)! is the factorial of n-r, which is the product of all positive integers up to n-r

For example, the number of permutations of 5 objects taken 3 at a time is:

$$P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60$$

This means that there are 60 different ways to arrange 5 objects taken 3 at a time.

Combinations

A combination is a selection of objects from a set without regard to order. The number of combinations of a set of n objects taken r at a time is given by the formula:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

where:

• C(n, r) is the number of combinations of n objects taken r at a time
• n! is the factorial of n, which is the product of all positive integers up to n
• r! is the factorial of r, which is the product of all positive integers up to r
• (n-r)! is the factorial of n-r, which is the product of all positive integers up to n-r

For example, the number of combinations of 5 objects taken 3 at a time is:

$$C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{120}{6} = 20$$

This means that there are 20 different ways to select 3 objects from a set of 5 objects without regard to order.

Example

Suppose you have a set of 5 letters: A, B, C, D, and E. The number of permutations of these letters taken 3 at a time is:

$$P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60$$

This means that there are 60 different ways to arrange these letters taken 3 at a time. Some examples of these permutations include:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

The number of combinations of these letters taken 3 at a time is:

$$C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{120}{6} = 20$$

This means that there are 20 different ways to select 3 letters from this set without regard to order. Some examples of these combinations include:

• ABC
• ABD
• ABE
• ACD
• ACE
• ADE
• BCD
• BCE
• BDE
• CDE

What are the real-life examples of permutations and combinations?

Permutations and combinations are two fundamental concepts in mathematics that deal with the arrangement and selection of elements from a set. They find numerous applications in various real-life scenarios. Here are some examples to illustrate their usage:

Permutations:

1. Ordering of Items: Suppose you have a bookshelf with 10 books. The number of ways you can arrange these books on the shelf is a permutation. There are 10! (10 factorial) possible permutations, which is 3,628,800 ways.

2. Sports Teams: In a sports league with 12 teams, the number of ways to determine the winner, runner-up, and third place is a permutation. There are 12! / (12-3)! = 1320 possible permutations.

3. Passwords: When creating a password, you can choose from a set of characters (e.g., letters, numbers, symbols). The number of possible passwords of length 8 is a permutation. If there are 95 available characters, there are 95^8 possible passwords.

Combinations:

1. Selecting Committee Members: Suppose you need to select a committee of 5 members from a group of 10 people. The number of ways to do this is a combination. There are 10C5 = 252 possible combinations.

2. Lottery: In a lottery where you need to select 6 numbers from a set of 49, the number of possible winning combinations is a combination. There are 49C6 = 13,983,816 possible combinations.

3. Handshakes: At a party with 10 people, the number of possible handshakes between each pair of people is a combination. There are 10C2 = 45 possible handshakes.

These examples demonstrate how permutations and combinations are used in various real-life situations to determine the number of possible arrangements or selections from a given set. Understanding these concepts is essential in fields such as probability, statistics, computer science, and many others.

Write the relation between permutations and combinations.

Relation between Permutations and Combinations

Permutations and combinations are two fundamental concepts in combinatorics, the branch of mathematics that deals with the selection and arrangement of objects. While both concepts involve selecting objects from a set, they differ in the way the objects are arranged.

Permutations

A permutation is an ordered arrangement of objects. The order in which the objects are arranged matters. For example, if we have the letters A, B, and C, there are six possible permutations of these letters:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

In general, the number of permutations of n objects is given by the formula:

$$P(n) = n!$$

where n! (read as “n factorial”) is the product of all positive integers up to n. For example, 3! = 3 x 2 x 1 = 6.

Combinations

A combination is a selection of objects without regard to order. The order in which the objects are selected does not matter. For example, if we have the letters A, B, and C, there are three possible combinations of these letters:

• AB
• AC
• BC

In general, the number of combinations of n objects taken r at a time is given by the formula:

$$C(n, r) = \frac{P(n)}{P(r)P(n-r)}$$

where P(n) is the number of permutations of n objects, P(r) is the number of permutations of r objects, and P(n-r) is the number of permutations of n-r objects.

For example, the number of combinations of 5 objects taken 3 at a time is:

$$C(5, 3) = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10$$

Relationship between Permutations and Combinations

The relationship between permutations and combinations can be seen from the following formula:

$$C(n, r) = \frac{P(n)}{P(r)P(n-r)}$$

This formula shows that the number of combinations of n objects taken r at a time is equal to the number of permutations of n objects divided by the product of the number of permutations of r objects and the number of permutations of n-r objects.

In other words, the number of ways to select r objects from a set of n objects without regard to order is equal to the number of ways to arrange the r objects in a specific order divided by the number of ways to arrange the remaining n-r objects in a specific order.

Example

Suppose we have a group of 10 people and we want to select a committee of 3 people. There are 10C3 = 120 ways to select the committee. If we want to arrange the committee members in a specific order, there are 3! = 6 ways to do so. Therefore, there are 120 x 6 = 720 ways to select and arrange the committee members.

Give the applications of permutation and combination in mathematics.

Permutations and combinations are two fundamental concepts in mathematics that deal with the arrangement and selection of elements from a set. They have numerous applications in various fields, including probability, statistics, computer science, and cryptography. Here are some detailed explanations and examples of their applications:

1. Permutations:

Permutations refer to the arrangement of elements in a specific order. The number of permutations of n distinct objects taken r at a time is given by the formula:

$$P(n, r) = n! / (n - r)!$$

Applications:

• Counting Arrangements: Permutations are used to count the number of possible arrangements of objects in a specific order. For example, if you have 5 books to arrange on a shelf, the number of ways you can do so is given by P(5, 5) = 5! = 120.

• Password Combinations: Permutations are used to generate secure passwords. For instance, if you create a 4-digit PIN using the digits 0 to 9, the number of possible PINs is given by P(10, 4) = 5040.

• Sports Scheduling: Permutations are used to create schedules for sports tournaments, ensuring that each team plays against every other team exactly once.

2. Combinations:

Combinations refer to the selection of elements without regard to their order. The number of combinations of n distinct objects taken r at a time is given by the formula:

$$C(n, r) = n! / (r! (n - r)!)$$

Applications:

• Selecting Committees: Combinations are used to select a committee of r members from a group of n people. For example, if you need to choose a 3-person committee from a group of 10 people, the number of possible committees is given by C(10, 3) = 120.

• Lottery Drawings: Combinations are used in lottery drawings to determine the winning combinations. For instance, in a lottery where you select 6 numbers from a pool of 49, the number of possible winning combinations is given by C(49, 6) = 13,983,816.

• Probability Calculations: Combinations are used in probability calculations to determine the likelihood of certain events occurring. For example, if you roll two dice, the probability of getting a sum of 7 is given by the number of ways to get a sum of 7 divided by the total number of possible outcomes. This can be calculated using combinations as follows:

Number of ways to get a sum of 7 = C(6, 2) = 15 Total number of possible outcomes = 6 * 6 = 36 Probability of getting a sum of 7 = 15/36 = 5/12

These are just a few examples of the many applications of permutations and combinations in mathematics. They provide powerful tools for counting, selecting, and arranging elements, and have a wide range of practical uses in various fields.

What is the factorial formula?

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

The factorial function has a number of important properties. First, 0! = 1. This is because the product of no numbers is defined to be 1. Second, n! = n × (n-1)!. This is because the factorial of n is the product of n and the factorial of n-1.

The factorial function is used in a variety of mathematical applications. For example, it is used to calculate the number of ways to arrange n objects in a row. It is also used to calculate the number of ways to select r objects from a set of n objects.

Here are some examples of how the factorial function is used:

• The number of ways to arrange 5 people in a row is 5! = 5 × 4 × 3 × 2 × 1 = 120.
• The number of ways to select 3 people from a group of 10 people is 10! / (3! × 7!) = 120.
• The probability of getting 6 heads in a row when flipping a coin is (1/2)^6 = 1/64.

The factorial function is a powerful tool that can be used to solve a variety of mathematical problems. It is important to understand the properties of the factorial function in order to use it effectively.

What does nCr represent?

What does nCr represent?

In combinatorics, nCr represents the number of ways to choose r elements from a set of n elements, where order does not matter. It is also known as the binomial coefficient.

The formula for nCr is:

$$nCr = \frac{n!}{r!(n-r)!}$$

where:

• n is the total number of elements in the set
• r is the number of elements to choose
• ! denotes the factorial function

Example:

Suppose you have a set of 5 people and you want to choose 3 of them to form a committee. There are 10 possible ways to do this:

• ABC
• ABD
• ABE
• ACF
• ACD
• ACE
• BCF
• BCD
• BCE
• CDE

Each of these combinations represents a different way to choose 3 people from the set of 5.

Applications of nCr

nCr is used in a variety of applications, including:

• Counting the number of possible outcomes in a probability experiment
• Selecting a random sample from a population
• Designing experiments
• Analyzing data

Additional examples

Here are some additional examples of how nCr is used:

• A company has 10 employees and wants to select 3 of them to attend a conference. There are 120 possible ways to do this.
• A survey is conducted of 100 people and 20 of them are found to be smokers. The probability of selecting a smoker from the sample is 20/100 = 0.2.
• A drug is tested on 100 patients and 60 of them are found to be cured. The efficacy of the drug is 60/100 = 0.6.

nCr is a powerful tool that can be used to solve a variety of problems in combinatorics and probability.