I have 4 variables.
$A= l{nq\choose l} {n(1-q)\choose n(1-q)-np+l}$
$B=(np-l){nq\choose l} {n(1-q)\choose n(1-q)-np+l}$
$C=(nq-l){nq\choose l} {n(1-q)\choose n(1-q)-np+l}$
$D=(n(1-q)-np+l){nq\choose l} {n(1-q)\choose n(1-q)-np+l}$
$l$ goes from $0$ to $np$, $p$ and $q$ are probabilities. So for example, $p=0.2, q=0.6, n=15$, we have $np=3, nq=9,n(1-q)=6,n(1-p)=12$.Also $np<nq$. When $np,nq,n(1-p), n(1-q)$ are fractions they are rounded off to their nearest whole number.
How can we optimise $A,B,C,D$ with respect to $l$ such that they try to attain their maximum value? One way I thought of was to maximise the product ABCD, find the $l$ for which $ABCD$ is maximum. As an example I tried for $n=15, p =0.2, q=0.6$ case.
I found that this maximum is occurs at $l=1.815$, but this is just by using Mathematica. How can I do this algebraically? I see that this $l=1.815$ is close to $npq$, Basically how can I get a general result for the $l$ corresponding to the maximum value of the $ABCD$ product in terms of n,p and q?