How many solutions have: $|X_1|+ \ldots +|X_n| = k$

Question

$k$,$n$ are both Natural numbers, $k\geq n$

$X_1,\cdots ,X_n$ are Integers. $|X_i|\geq 1$.

I have to find how many solutions have $|X_1|+\dots+|X_n|=k$.

I started with:

$|X_1+\cdots +X_n|\leq|X_1|+\cdots +|X_n|=k$

$|X_1+\cdots +X_n|+X_n+1=k$

But I'm not sure how to continue. Please help.

Answer 1

Assume, first, that all of them are positive, then is a stars and bars argument that gives $\binom{k-1}{n-1}$(Do you see how?) now, as there is no difference in between choosing negative or positive numbers, you have $2$ choices for each summand, and hence there should be $\binom{k-1}{n-1}2^n$ of those solutions.