Optimisation Problem over binomial coefficients

Question

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?

Answer 1

p = 0.2; q = 0.6; n = 15;
a = l*Binomial[n q, l] Binomial[n (1 - q), n (1 - q) - n p + l];
b = (n p - l) Binomial[n q, l] Binomial[n (1 - q), n (1 - q) - n p + l];
c = (n q - l) Binomial[n q, l] Binomial[n (1 - q), n (1 - q) - n p + l];
d = (n (1 - q) - n p + l) Binomial[n q, l] Binomial[n (1 - q), n (1 - q) - n p + l];

(* maximize for each individually *)
NMaximize[{a, 0 < l <= n*p}, l]
NMaximize[{b, 0 < l <= n*p}, l]
NMaximize[{c, 0 < l <= n*p}, l]
NMaximize[{d, 0 < l <= n*p}, l]

(* maximize the product abcd *)
NMaximize[{a*b*c*d, 0 < l <= n*p}, l]
(* maximize the sum a+b+c+d *)
NMaximize[{a + b + c + d, 0 < l <= n*p}, l]
(* maximize the minimum *)
NMaximize[{Min[a, b, c, d], 0 < l <= n*p}, l]