Compute the number of positive integer solutions to the equation $x_{1} + x_{2} + x_{3} = 25$. Where 1 $\leq x_{1}, x_{2}, x_{3}$ $\leq$ 15. The book I am using for my Combinatorics course is Combinatorics:Topics, Techniques, and Algorithms.
Compute the number of positive integer solutions to the equation $x_{1} + x_{2} + x_{3} = 25$. Where 1 $\leq x_{1}, x_{2}, x_{3}$ $\leq$ 15.
This is my rough proof to this question. I was wondering if anybody can look over it and see if I made a mistake or if there is a simpler way of doing this problem. I want to thank you ahead of time it is greatly appreciated.So lets begin:
Proof:


One way to do this is to define $w=x+x^2+x^3+ \cdots +x^{15}$ and then cube $w$ and determine the coefficient of $x^{25}$ in the expanded version of $w^3.$ That coefficient comes out $168$ when I used maple. I don't know any "clever" way to get this without the aid of something like maple, and this approach would be quite a mess by hand. Each term of $w^3$ which contributes to the $x^{25}$ term represents one way to get a sum of 25 using three summands from 1 to 15.
