Number of different sums with k numbers from {1, 5, 10, 50} 
Say we have $k$ numbers, each of which belongs to the set $S = \{1, 5, 10, 50\}$
  How many different sums can be created by adding these numbers?

If $k = 1$, the are four different sums.
Also, if $k = 2$, there are ten: 
$$\begin{align} 1 + 1 = 2  \quad 1 + 5 &= 6 \quad 1 + 10 = 11 \\ 1 + 50 = 51 \quad 5 + 5 &= 10 \quad 5 + 10 = 15 \\ 5 + 50 = 55 \quad  10 + 10 &= 20 \quad 10 + 50 = 60 \\ 50 + 50 &= 100 \quad \end{align}$$

At least for small values of $k$, this answer will be tractable. Your answer is the number of terms in the expansion of $(x+x^{5}+x^{10}+x^{50})^k$ which can be found using a computer algebra system.
