Hello, nbukhar!

1. Three positive integers can be used to form a trio.

For example (1,2,2) is a trio whose sum is 1 + 2 + 2 = 5

and (2,1,2) is a different trio whose sum is also 5.

How many trios can you find with a sum of 5? The very least you can do is __list__ the trios. .Did you try?

There are **six**: .(1,2,2), (2,1,2), (2,2,1), (1,1,3), (1,3,1), (3,1,1)

I assume "Investigate further" means to try to generalize the problem.

For example, how trios have a sum of 7?

Consider a 7-foot board, marked at intervals of one foot.

We want to cut it into three pieces. Code:

* - * - * - * - * - * - * - *
| | | | | | | |
* - * - * - * - * - * - * - *
↑ ↑ ↑ ↑ ↑ ↑

There are 6 places to make the cuts; we will choose 2 of them.

. . Hence, there are: C(6,2) = 15 choices.

Therefore, there are 15 trios with a sum of 7.

In general, for a whole number *n* __>__ 3, there are: .C(n-1, 2) trios.