Summation over nCr

"I'm a fan of anything that tries to replace actual human contact." - Sheldon. After years of hard work, Sheldon was finally able to develop a formula which would diminish the real human contact. He found $k$ integers $n_1,n_2...n_k$ . Also he found that if he could minimize the value of $m$ such that $\sum_{i=1}^k$$n$$i$$C$$m$$i$ is even, where $m$ = $\sum_{i=1}^k$$m_i$, he would finish the rea