Question

Each of the digits 1, 1, 2, 3, 3 and 4 is written on a separate card. The six cards are then laid out in a row to form a 6-digit number. How many of these 6-digit numbers are divisible by 4?

Answer 1

In order to get the 6-digit number divisible by 4

The last two digits must be divisible by 4

∴ The last two digits should be 12 or 24 or 32

				24
1	2	3	4	5

Let the last box be filled with 24.

The remaining 4 boxes can be filled with the remaining digits

1, 1, 3, 3 in `(4!)/(2! xx 2!)` ways.

				12
1	2	3	4	5

Let the last box be filled with 12.

The remaining 4 boxes can be filled with the remaining digits.

1, 3, 3, 4 in `(4!)/(2!)` ways

				32
1	2	3	4	5

Let the last box be filled with 32.

The remaining 4 boxes can be filled with the remaining digits

The total number of 6 digit numbers which are divisible by 4 is

= `(4!)/(2! xx 2!) + (4!)/(2!) + (4!)/(2!)`

= `(1 xx 2 xx 3 xx 4)/(1 xx 2 xx 1 xx 2) + (1 xx 2 xx 3 xx 4)/(1 xx 2) + (1 xx 2 xx 3 xx 4)/(1 xx 2)`

= 6 + 12 + 12

= 30

∴ Required number of 6-digit numbers = 30