Maximization of $\sum_{\mathbf{c}_m} \prod_k \left[ (1-p_{k})^{N} p_{k}^{c_{mk}} \right]$

Question

Given a matrix $\mathbf{C}$ ($M \times K$) that contains $M$ vectors of $K$ non-negative integers, I have the following probability distribution.

\begin{align} \sum_{\mathbf{c}_m} \prod_k \left[ (1-p_{k})^{N} p_{k}^{c_{mk}} \right] \end{align}

I would like to:

• Maximize it with respect to $\mathbf{p}=(p_1,...,p_k)$ when $\mathbf{C}$ has this kind of structure (note that every $\mathbf{c}_m$ sums up to the same number)

       [k1] [k2]
[m1,]    0    3
[m2,]    1    2
[m3,]    1    2
[m4,]    2    1
[m5,]    2    1
[m6,]    3    0

I derivate with respect to $p_k$ and equal to zero and I obtain:

\begin{align} N\frac{p_k}{1-p_k} = \frac {\sum_{\mathbf{c}_m} c_{mk}\prod_{k} p_{k}^{c_{mk}} } {{\sum_{\mathbf{c}_m}} \prod_{k} p_{k}^{c_{mk}} } \end{align}

Any ideas on how to solve this?