Say we have $n$ sets, each set containing $x_k$ elements that can be distinguished. Now, we choose $m$ elements from those sets and number each element sequentially. In each position, an element is placed, but you can't place two elements adjacently if they come from the same set.
\begin{equation*} \text{If }N\text{ is the number of ways that can be done, how can we find }N\text{?} \end{equation*}
\begin{equation*} \text{Is there a general closed form for this problem?} \end{equation*}
$\textbf{Example}$: We randomly shuffle a deck of cards (13 cards in each four suits) and draw them ordering in a line. How many ways can we do that such that there are no two adjacent cards that belong to the same suit?
I'm aware that this has been answered:
\begin{equation*} N=\int \limits _0 ^{\infty } e^{-x}(q_{13}(x))^{4}\text{d} x, \quad q_k=\sum _{i=1} ^k \binom{k-1}{i-1} \frac{x^i(-1)^{i-k}}{i!} \end{equation*}
$\textbf{Rewording my question with an example}$: What if the suit of spades had 20 cards, diamonds 10, clubs 15 and hearts 25. We'd like to do that process with 10 cards from any suit. How many ways could we do that?
