Combination of natural numbers (1-10) in a triangle

Question

I have the numbers 1..10 and want to arrange them in a billiard triangle such that they add up to the value on the right hand side thus. I can only use each number once.

Is there a formula for working out the cominations? Any hints about proving it?

Alternatively: Knowing that ? must be 10 by Gauss' formula I have a list of combinations for each line:

line 2:

9,4
8,5
7,6

Line 3:

9,5,1
8,6,1
7,6,1
9,4,2
8,5,2
8,4,3

Line 4:

9,5,2,1
9,4,3,1
8,6,2,1
8,5,3,1
$\cdots$

Answer 1

You've already got everything you need. You know the $10$ goes at the top; and you have disjoint lists of balls for the second and third rows, e.g. $9, 4$ and $8,6,1$. Then just put the remaining balls $2,3,5,7$ in the last row.