Combinatorics with constraints

Inclusion-Exclusion Combinatorics

function factorial(n){
	return (n < 2) ? 1 : n * factorial(n-1);
}

function C(n,k){
	return factorial(n) / (factorial(k) * factorial(n-k));
}

function maxPossibleConfigurations(a, b, c, n){
	var args= [].slice.call(arguments);
	if(args.length < 4) throw new Error("invalid arguments ["+args.join(", ")+"]");
	if(a+b+c<n) return 0;
	return C(2+n, 2) - C(2+n-a, 2) - C(2+n-b, 2) - C(2+n-c, 2) + 2*C(2+n-a-b, 2) + 2*C(2+n-b-c, 2) + 2*C(2+n-a-c, 2);
}