A five digit number is formed by using the digits {1, 2, 3, 4, 5, 6}. Find the total number of such numbers which are divisible by 55.

A. 9

B. 10

C. 11

D. 12

JEE Mains 2021 Maths Paper

Solution:

It is given that the five-digit number is to be formed with digits 1, 2, 3, 4, 5, 6

The five-digit number should be divisible by 55, which means it should be divisible by 11 and 5.

To be divisible by 5, the last digit of the number should be 5.

Let the five-digit number be abcd5.

Now, in order to be divisible by 11,

|Sum of digits at odd places - Sum of digits at even places| = divisible by 11

|(a+c+5) - (b+d)| = divisible by 11

Case 1:

(a+c+5) - (b+d) = 0

(a+c) = (b+d) - 5

This is possible for the values of (b, d) = (6, 4) and (a, c) = (2, 3)

There will be 4 possible cases from these digits.

Case 2:

(a+c+5) - (b+d) = 11

(a+c) = (b+d) + 6

This is possible for the values of (b, d) = (1, 2) and (a, c) = (3, 6)

There will be 4 possible cases from these digits.

Case 3:

(a+c+5) - (b+d) = 11

(a+c) = (b+d) + 6

This is possible for the values of (b, d) = (1, 3) and (a, c) = (6, 4)

There will be 4 possible cases from these digits.

Hence, there will be a total of 12 possible cases.

Hence, (d) is the correct answer.

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