Let there be 4 integers $w,x,y,n$ belonging to $ℤ+$ such that $w,x,y \geq n$ and $n \gt 2$. Find the value of $w$ in terms of $x, y, n$ or prove that it's impossible if $ x \neq y $ and
$\frac{w!}{(w-n)!} = \frac{x!}{(x-n)!} + \frac{y!}{(y-n)!}$
I faced this problem while trying to solve a probability problem, if I have a box called $A$ with $x+y$ balls and a box $B$ with $x$ white balls and $y$ black balls, find the number of white balls in $A$ such that the probability you pick out $n$ balls either all white or black from $B$ is equal to the probability you pick out $n$ white balls from $A$. It boiled down to the equation I wrote above, and after some analysis (the consecutive nature of the integers etc.) it seemed impossible for $n$ to be greater than 2. I tried proving it, and was thinking in the direction of prime factors & mod. Also, I noticed the equation is equivalent to $^w P_n$ = $^x P_n$ + $^y P_n$ as well, but thinking in that direction didn't really prove fruitful.