Question

The number of seven digit integer is to be formed using only 1, 2 &3 such that the sum of all the digits is 10 . so how many such seven digit number is possible ?

Answer 1

I got `42` different numbers.

Explanation:

We start by listing the combinations of numbers that give us a digit sum of `10`.

We can have

`1, 1, 1, 1, 1, 3, 2`

And the number of different arrangements here is `(7!)/(5!) = 42`.

Is the above sequence the only possible?

Note that you need a certain number of `1`'s to make the number of digits `7` and the sum `10`, because the number `2222222` has a digit sum of `14`, for example. If we try other sequences, such as

`1, 1, 1, 1, 1, 1, 1, 3`

We either get sequences that are more or less than `7` terms or that have a sum other than `10`. I'm not sure how to prove that the above sequence is the only one possible, so I'll leave that to other contributors.

Hopefully this helps!