Does this sum of products of binomial coefficients have a simple closed form?

Question

Let $c,m,k$ be positive integers. Is there a simple closed form for the following sum? $$ \sum_{i=1}^{c-1} (-1)^i {c \choose i} {im \choose k} $$ Mathematica finds nothing, and Maxima's implementation of Gosper's algorithm says that the summand isn't hypergeometric in $i$.

Answer 1

$$\sum_{i=0}^{c}(-1)^i\binom{c}{i}\binom{im}{k}=[x^k]\sum_{i=0}^{c}(-1)^i\binom{c}{i}(1+x)^{mi}=[x^k]\left(1-(1+x)^m\right)^c.$$