kill(all)$

van=facebook.com/LearningCalculus$

f(a,b):=binomial(a,b)$

f(a,b) = makefact(f(a,b));

f(a,b)*f(b,c)*f(a,d)=f(a,c)^2*f(c,d);

1/makefact(%);

(a-b)!*(b-c)!*c!*(a-d)!*d!=(a-c)!^2*c!*(c-d)!*d!;

(a-b)!*(b-c)!*(a-d)!=(a-c)!^2*(c-d)!;

assertion:subst(u,a-c,subst(v,b-c,%));

question: subst(u,a-c,subst(v,b-c,makefact(f(a-c,b-c))));

(a-b)!*(a-d)!*v!  /(a-b)!/(c-d)!/v!/u!=
(c-d)!*       u!^2/(a-b)!/(c-d)!/v!/u!;

f(a-c,b-c)=subst(a-c,u,rhs(rhs(%)=lhs(%)));

f(x,y):=abs(x)!*((abs(x)-abs(y))!*abs(y)!);