I am having trouble with some combinatorial question. Its not my field and the question is difficult for me.
I've asked related question before, combinatorial question (sum of numbers) but it turned out that I formulated it wrong. But it was interesting problem any way. Thank you very much for the help!
My real question is: Given numbers $r$ and $m$. Let $m_1,..., m_{{2r}}$ be numbers such that $m_i \in \{0, 1, ..., 2m\}$ and $\sum_{i=1}^{2r}m_i=2m$.
Find number of ways choosing $m_1,...,m_{2r}$, such that sum of any $r$ of them will be odd.
I was trying to calculate number of repetitions (order is not important) and then just subtract it from the result in the question 'combinatorial question (sum of numbers)'. But it seems I have to use different procedure.
Thank you.