I am having some trouble understanding if the following expression can be simplified
\begin{equation} \binom{m}{b, c'-b, c-b, m+b-(c'+c)} P_s^{m+2b-(c'+c)} (1-P_s)^{((c'+c)-2b)} \end{equation}
where $0<P_s<1$.
Notice here that the probabilities of picking the first and last element in the multinomial coefficient are the same ($P_s$), and the probabilities of the second and third are the same ($1-P_s$). Given this symmetry, is there any way to simplify the coefficient?
Thanks, James
The first check you need to make is that all the coefficients add up to $m$, as they must; which they do.
Collect the coefficients of the $2{nd}$ and $3{rd}$ terms: $(c+c'-2b)$
Cleaning up the clutter of symbols, I replace $(c+c'-2b)\;to\; a$ and $P_s\; to\; p$,
Then the probabilities part of the expression will simply be
$\Large p^{m-a}\cdot(1-p)^a$
ADDED
If it is the multinomial coefficient you were talking about, there's nothing, really,you can do, except, maybe, put parts with the same probability in adjoining manner, viz
$$\binom{m}{b,\;\, m+b-(c'+c),\;\,c'-b,\;\,c-b}$$