Permutations and Combinations Exercise 01
Question:
How many 5 digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Answer:
 There are 10 digits that can be used (0 to 9).
 The first two digits are 67, so there are 8 digits left to choose from.
 Since no digit can appear more than once, each of the remaining 8 digits can be used only once.
 This means that there are 8 x 7 x 6 x 5 x 4 = 16,800 possible 5 digit telephone numbers that can be constructed.
Question:
Given 5 flags of different colours, how many different signals can be generated if each signal require use of 2 flags one below the other?
Answer:

First, determine how many different colour combinations are possible with two flags. This can be done by multiplying the number of colours (5) by itself (5 x 5 = 25).

Next, determine how many different signals can be generated with each colour combination. This can be done by multiplying the number of colour combinations (25) by the number of ways to arrange the flags (2). Therefore, the total number of signals that can be generated is 25 x 2 = 50.
Question:
How many 4letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Answer:
Answer:
Step 1: Determine the number of letters in the code. Answer: 4
Step 2: Determine the number of letters in the alphabet. Answer: 10
Step 3: Calculate the number of possible codes by using the formula: n^r, where n is the number of letters in the alphabet and r is the number of letters in the code. Answer: 10^4 = 10,000
Question:
How many 3 digit number can be formed from the digit 1,2,3,4 and 5 assuming that (i) repetition of the digit is allowed? (ii) repetition of the digits is not allowed?
Answer:
(i) Repetition of the digits is allowed:
Since repetition of the digits is allowed, each 3digit number can be formed by selecting any one digit from the given 5 digits, 3 times.
Therefore, the total number of 3digit numbers that can be formed = 5 x 5 x 5 = 125
(ii) Repetition of the digits is not allowed:
Since repetition of the digits is not allowed, each 3digit number can be formed by selecting any one digit from the given 5 digits, 3 times without repetition.
Therefore, the total number of 3digit numbers that can be formed = 5 x 4 x 3 = 60
Question:
How many 3digit even numbers can be formed from the digits 1,2,3,4,5,6 if the digits can be repeated?
Answer:
Answer:
 First, we need to determine how many different numbers can be formed using the digits 1,2,3,4,5,6. This can be done by calculating 6 x 6 x 6 = 216.
 Next, we need to determine how many of these numbers are even. To do this, we can count the number of even numbers that can be formed from the digits 1,2,3,4,5,6. There are 2 even numbers (2 and 4) and 4 odd numbers (1,3,5,6). Therefore, the number of even numbers that can be formed is 2 x 2 x 2 = 8.
 Finally, we need to determine how many 3digit even numbers can be formed from the digits 1,2,3,4,5,6 if the digits can be repeated. Since each digit can be repeated, the number of 3digit even numbers that can be formed is 8 x 8 x 8 = 512.
Question:
How many 5digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Answer:
Answer:

There are 10 possible digits that can be used (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

Since each number starts with 67, there are 8 digits remaining to be filled (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

Since no digit appears more than once, each of the 8 remaining digits must be different.

Therefore, the total number of 5digit telephone numbers that can be constructed is 8 x 7 x 6 x 5 x 4 = 16,800.