Putting $n$ balls in $m$ boxes and in each box is even number of balls

Question

I'm trying to count number of ways to put $n$ identical balls in $m$ identical boxes that in each box is even number of balls. I figure to put balls in pairs so in each box will be even number. Is this correct $\binom{n/2}{m}$?

Answer 1

If balls are indistinguishable and boxes are indistinguishable, you have a partition of $n/2$ ball-pairs into (up to) $m$ parts, with dependency on whether zero is an allowed number of balls in a box. No zeroes allowed would be $p_m(n/2)$; allowing zeroes would simply map to the same problem with more ball-pairs, $p_m(n/2+m)$ (to provide at least one ball-pair in each box).