Suppose $\displaystyle k,n \geq 0$. How many $\displaystyle k$-tuples $\displaystyle (x_1,\ldots,x_k)$ are there for which each $\displaystyle x_i \geq 0$ and $\displaystyle x_1+\ldots+x_k \leq n$?
introduce anouter variable a≥1, and let $\displaystyle x_1+...+x_k+a=n$ Clearly, $\displaystyle x_1+...+x_k<n$ Now, we solve this equality. Subtract one from both sides, and let u=a-1. Then u≥0--like the other variables. So now, we have n-1 on the RHS and k+1=>k + signs. So, there are (n-1+k, n-1) solutions to the inequality.