Permutations and Combinations Exercise 04
Question:
A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
Answer:
Answer: Step 1: Calculate the total number of balls in the bag = 5 black balls + 6 red balls = 11 balls
Step 2: Calculate the total number of ways in which 2 black and 3 red balls can be selected = (5C2) × (6C3)
Step 3: Substitute the values in the formula = (5C2) × (6C3) = (5! / (2! × 3!)) × (6! / (3! × 3!))
Step 4: Simplify the expression = (5! / (2! × 3!)) × (6! / (3! × 3!)) = 10 × 20 = 200
Step 5: The number of ways in which 2 black and 3 red balls can be selected is 200.
Question:
Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
Answer:
Answer: Step 1: Calculate the total number of balls = 6 red balls + 5 white balls + 5 blue balls = 16 balls
Step 2: Calculate the number of ways of selecting 3 balls of each colour = 6C3 x 5C3 x 5C3 = 120
Step 3: Calculate the number of ways of selecting 9 balls from 16 balls = 16C9 = 11,639,040
Step 4: Calculate the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls = 11,639,040 x 120 = 1,396,688,800
Question:
Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
Answer:

Count the number of aces in the deck of cards: 52 cards, 4 aces.

Calculate the number of combinations of 5 cards out of 52 cards: 52C5 = 2,598,960.

Multiply the number of combinations by the number of aces in the deck: 2,598,960 x 4 = 10,395,840.

The number of 5 card combinations out of a deck of 52 cards with exactly one ace in each combination is 10,395,840.
Question:
In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
Answer:
Step 1: Identify the total number of boys and girls. There are 5 boys and 4 girls.
Step 2: Calculate the total number of possible combinations. This can be calculated using the formula nCr, where n = total number of boys and girls and r = number of boys and girls to be selected.
In this case, n = 9 (5 boys + 4 girls) and r = 3 (3 boys + 3 girls).
Therefore, the total number of possible combinations is 9C3 = 84.
Question:
In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?
Answer:
 There are 17 players from which to choose a cricket team of 11.
 Of the 11 players, exactly 4 must be bowlers.
 There are 5 players who can bowl, so there are 5 possible choices for the 4 bowlers.
 For each selection of 4 bowlers, there are (174=13) players left to choose from for the remaining 7 positions.
 There are (13 choose 7) ways to select the remaining 7 players from the 13.
 Therefore, the total number of ways to select a cricket team of 11 from 17 players in which only 5 players can bowl and each team must have exactly 4 bowlers is (5 x (13 choose 7)) = 5 x 1716 = 8580.
Question:
How many chords can be drawn through 21 points on a circle ? A 301 B 210 C 111 D 220
Answer:
Answer: D 220
Question:
In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?
Answer:
Step 1: Count the total number of courses available to the student, which is 9 courses.
Step 2: Subtract the number of compulsory courses (2) from the total number of courses available (9). This leaves the student with 7 courses to choose from.
Step 3: Calculate the number of possible combinations of 5 courses from the 7 available courses. This can be done using the formula for combinations, which is: nCr = n!/(r!(nr)!).
Step 4: Plug the values into the formula. In this case, n = 7 and r = 5. This gives us 7!/(5!(75)!), which simplifies to 7!/5!.
Step 5: Calculate the result. 7!/5! = 7654321/54321 = 7*6 = 42.
Therefore, the student can choose a programme of 5 courses from the 9 courses available in 42 different ways.