Calculating the probability of combinations with unequal probabilities for each element
The value you want is the coefficient of $x^n$ in the polynomial $\displaystyle\prod_{i=1}^{n}(1+p_ix)$. You can multiply the factors one at a time and discard terms with $x^m$ for $m>k$. This will save some work if $k$ is much smaller than $n$. You can also use the binomial theorem to evaluate the product of repeated factors, if any of your $p_i$ values are equal. In your example, $p$ is $0.0036,$ not $0.003675$. $$(1+0.5) (1+0.3x) (1+0.15x) (1+0.05 x)=1 + 1. x + 0.3175 x^2 + 0.036 x^3 + 0.001125 x^4$$