Sum of binomial coefficients problem

Question

Let $(x_n)$ be a sequence of nonnegative integers such that, for any integer $n \geq 2$ with $0 \leq x_k \leq n$, $$\sum_{k=0}^n \binom{n}{x_k} = 2^n$$ How to show that $$\sum_{k=0}^n \frac{n+1-2x_k}{n+1-x_k}\binom{n}{x_k} = 1$$ ?

Answer 1

This is false. If it were true, we should be able to replace any $x_k$ by $n-x_k$; this doesn’t change the first sum and thus shouldn’t change the second sum. But it does, as you can see e.g. by replacing $x_k=0$ by $x_k=n$.