Question

Find the number of 4 digit numbers that can be formed using the digits 0, 2, 4 and 5 which are not divisible by 5 if repetition is not allowed.
[a] 10
[b] 8
[c] 6
[d] 4

Answer 1

Hint: A number that is divisible by 5 has either 0 or 5 at its units place. Hence those numbers which are not divisible by 5 formed using the digits 0, 2, 4 and 5 will have either 2 or 4 at their units place Also a 4 digit number cannot start with 0. Use the fundamental theorem of counting to find the total number of possible numbers.

Before solving the question, we need to understand what fundamental theorem of counting states. According to the fundamental theorem of counting If a task can be done in m ways and another task can be done in n ways then the total number of ways in which both tasks can be done is mn and the total number of ways in which either of the tasks can be done is m+n.

Complete step-by-step answer:

In the case of creating a 4 digit number, we need to fill all the 4 places. So the total number of ways in which the number can be created will be the product of the times each place is filled.

[a] we have 2 cases

Case I: The number starts with 5.

For units place, we have 2 choices viz 2 and 4 because 0 cannot be at units place.
For tens place, we also have 2 choices since the numbers placed at thousands place, and the units place cannot be used again as repetition is not allowed.
For hundreds please we are left with only one choice.
Hence the total number of ways in which the number can be created so that it is not divisible by 5 and the number starts with 5 when repetition is not allowed is equal to $2\times 2\times 1=4$

Case II: The number does not start with 5

In this case, we have 2 choices at thousands place because 0 and 5 both cannot be used and in units place, we are left with only one choice because 0, 5 and the number at thousands place cannot be used. For tens places, we have 2 choices 0 and 5, and for hundreds places, we will have only one choice.
Hence the total number of ways in which the number can be created so that it is not divisible by 5 and the number does not start with 5 when repetition is not allowed is equal to $2\times 1\times 2\times 1=4$

Hence the total number of ways = 4 + 4 = 8 {Because either we will form a Case I number or a
Case II number.}

Hence the total number of 4 digit numbers formed using the digits 0, 2, 4 and 5 when repetition is not allowed which are not divisible by 5 = 8

Hence option [b] is correct.

Note:
[1] When A or B needs to be done then the total number of ways = n(A)+n(B)
When both A and B need to be done then the total number of ways = n(A) n(B)
You can remember this as “or” = + and “and” = *, just like in Boolean algebraic representation

[2] When solving such types of questions, try first filling those places which have the most restrictions.This makes the question easier to solve. In the above question, the most restrictive places were the units place and the thousands place. So, we started by first filling those places.