# Sum over a product of binomial coefficients related to a collision problem

I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:

$$R\left(n,m,j\right)=\sum_{k=0}^{n}a^{k}b^{n-k}c^{j-k}d^{k+m-j}\binom{j}{k}\binom{n+m-j}{n-k}\binom{n}{k}\binom{m}{j-k}$$

for positive real numbers $a,b,c,d\leq1$ and positive integers $n$, $m$, and $j$ with $j\leq m+n$.

The probability that the collision involved $n$ particles is given by $\left(1-\delta\right)\left(\delta^{n}\right)$ and the probability that collision involved $m$ particles is given by $\left(1-\delta\right)\left(\delta^{m}\right)$, for a positive real number $\delta<1$.

Then what is the probability of forming $j$ particles, $Pr\left(j\right)$? Is there a simple form for $Pr\left(j\right)$ that does not depend on $n$ or $m$?

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You are probably right. I have taken it out. I am not sure if the equation for Pr(j) is now correct. – GabrielM Feb 19 '14 at 20:41
If $k>j$ then the first binomial coefficient $\binom{j}{k}$ should be 0 – GabrielM Feb 19 '14 at 22:38
So you can put a $1_{[0,j]}(k)$ factor in there... and some more factors like that for the other binomial coefficients. – Bjørn Kjos-Hanssen Feb 19 '14 at 23:04