The number of five-digit numbers which are divisible by `3` that can be formed by using the digits `1,2,3,4,5,6,7,8` and `9`, when repetition of digits is allowed, is (a)`3^(10)` (b)`4.3^(8)` (c)`5.3^(8)` (d)`7.3^(8)`

Options

A. `3^(10)`
B. `4.3^(8)`
C. `5.3^(8)`
D. `7.3^(8)`

Answer

A

Explanation

`(a)`
`1^(st)` blank can be filled in `9` ways
`2^(nd)` blank can be filled in `9` ways (repetition is allowed)
`3^(rd)` blank can be filled in `9` ways
`4^(th)` blank can be filled in `9` ways
Now, we have to fill the `5^(th)` blank carefully such that the number is divisible by `3`.
Add the `4` numbers in the first `4` blanks.
If their sum is in the form `3n`, then fill the last blank by `3`, `6` or `9` so that the sum of all digits is divisible by `3`.
If their sum is in the form `3n+1`, than fill the last bank by `2`, `5` or `8`.
If their sum is in the form `3n+2`, than fill the last blank by `1`, `4` or `7`.
Therefore, in any case, the last blank can be filled in `3` ways only.