In how many ways can we list out a string of seven numbers (using only the numbers 1, 2, and 3) such that they sum to 10?
Let's first look at the number of ways we can have 7 digits with the number 1, 2, and 3 add up to 10:
In fact, there are no other combinations of numbers that will get us to 10.
Ok, so how many permutations can we make with these combinations of numbers?
There are 7 places the number 3 can go - so that is 7.
Once the 3 is placed, there are 6 places the 2 can go - so that is 6.
The rest of the numbers are 1's and so there is only 1 way to do the "filler" with the 1s.
So there are
We can place the three 2s across the seven places in many different ways. This is a combinations problem (we don't care which 2 ends up where), and so there are:
And so there are