Non-negative integral solutions of $X_1+X_2+X_3+X_4<n$

Question

The number of non-negative integral solutions of $X_1+X_2+X_3+X_4<n$ (where $n$ is a positive integer) is?

Answer 1

It' the same as the number of integral non-negative solutions to $X_1+X_2+X_3+X_4+X_5=n-1$, where $X_5$ is the difference, which is $\binom{n-1+5-1}{5-1}=\binom{n+3}{4}$

Answer 2

Start with $X_1 < n$ there are $n$ values $0,1,\cdots,n-1$ so call $S_1(n) = n$.

Now we can try $X_1 + X_2 < n$ there are a triangle of values so $S_2(n) = T(n)$.

The triangular numbers are the sum of the first $n$ numbers and in general we have $$S_k(n) = \sum_{i=0}^{n} S_{k-1}(i).$$

These are just the binomial coefficients.