# 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 ?

Apr 8, 2017

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!