Number of possible solutions for an equation

Solution 1:

Suppose that "natural number" doesn't include 0. Take 25 balls, and put a wall between any two. A partition $a+b+c=25$ is the same as a choice of two walls, and there are $\binom{24}{2} = 276$ of these.

If zeroes are allowed, then $(x+1)+(y+1)+(z+1) = 28$, and so the answer is $\binom{27}{2} = 351$.

Solution 2:

This is called the "partition function" of 25. See here:

http://en.wikipedia.org/wiki/Partition_function_%28number_theory%29

It's not an easy-to-compute-directly function - the best bet is simply writing a small script that counts solutions using a double loop.

Solution 3:

HINT $\rm\ \ (x,y,z)\ \to\ \{x,\ x+y\}\ $ bijects solutions with two elt subsets of $\{1,2,\cdots,24\} $