<s>[INST] 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: 
$$\begin{align} 1 + 1 = 2  \quad 1 + 5 &= 6 \quad 1 + 10 = 11 \\ 1 + 50 = 51 \quad 5 + 5 &= 10 \quad 5 + 10 = 15 \\ 5 + 50 = 55 \quad  10 + 10 &= 20 \quad 10 + 50 = 60 \\ 50 + 50 &= 100 \quad \end{align}$$
 [/INST] We can calculate the    number of solutions with some  algebra. We represent multiples of $\{1,5,10,50\}$ by  generating functions (in $x$) and count the number of occurrences in $y$.

We calculate the number of sums with $k$ members from $\{1,5,10,50\}$ as the coefficient of  $[y^k]$  evaluated at $x=1$:
  \begin{align*}
\left.\left([y^k]\frac{1}{1-xy}\frac{1}{1-x^5y}\frac{1}{1-x^{10}y}\frac{1}{1-x^{50}y}\right)\right|_{x=1}
\end{align*}

The series starts with
\begin{align*}
1&+(x+x^5+x^{10}+x^{50})y\\
&+(x^2+x^6+x^{10}+x^{11}+x^{15}+x^{20}+x^{51}+x^{55}+x^{60}+x^{100})y^2\\
&+\cdots
\end{align*}
which is evaluated at $x=1$
\begin{align*}
\color{blue}{1}+\color{blue}{4}y+\color{blue}{10}y^2+\cdots
\end{align*}
and with some help of Wolfram Alpha we find in accordance with the example of @saulspatz
\begin{align*}
\color{blue}{[x^{55}y^6]}\frac{1}{1-xy}\frac{1}{1-x^5y}\frac{1}{1-x^{10}y}\frac{1}{1-x^{50}y}\color{blue}{=2}
\end{align*}
</s>