# Questionbank: Permutation and Probability

One unit number theory and one unit Combinatorics makes a potent combination for fabulous questions!

## Counting 7 Digit Numbers

Q.5: A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?
1. 11
2. 12
3. 10
4. 22

Choice A. 11

## Detailed Solution

Number should be a multiple of 3 and 4. So, the sum of the digits should be a multiple of 3. WE can either have all seven digits as 3, or have three 2's and four 3's, or six 2's and a 3. (The number of 2's should be a multiple of 3).

For the number to be a multiple of 4, the last 2 digits should be 32. Now, let us combine these two.

All seven 3's - No possibility.

Three 2's and four 3's - The first 5 digits should have two 2's and three 3's in some order. No of possibilities = ${5! \over 3!2!}$ = 10

Six 2's and one 3 - The first 5 digits should all be 2's. So, there is only one number 2222232.

So, there are a total of 10 + 1 = 11 solutions.

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