2005 Putnam B4

For positive integers $m$ and $n$, let $f(m,n)$ denote the number of

$n$-tuples $(x_1,x_2,\dots,x_n)$ of integers such that

$|x_1| + |x_2| + \cdots + |x_n| \leq m$.

Show that $f(m,n) = f(n,m)$.

对于正整数 $m$ 和 $n$，令 $f(m,n)$ 表示

$n$ 个整数元组 $(x_1,x_2,\dots,x_n)$ 使得

$|x_1| + |x_2| + \cdots + |x_n| \leq m$。

证明 $f(m,n) = f(n,m)$。