# Is there a direct combinatorial interpretation of this (seemingly coincidental) result?

The original problem is as follows:

Find the probability $p$ that in a bridge game the players North, East, South, West have $a, b, c, d$ spades, respectively (which implies $a+b+c+d=13$).

And after simple counting, you'll get the answer, which can be rewritten as: $$p=\frac{\binom{13}{a}\,\binom{13}{b}\,\binom{13}{c}\,\binom{13}{d}}{\binom{52}{13}}$$

And it seems like this result can be directly obtained by interpreting the numerator and denominator respectively, but I'm stuck with how to do this. Any ideas?

• Don't think your answer is correct. Have a look at math.stackexchange.com/q/1416902/321264. – StubbornAtom Aug 19 '16 at 11:08
• @StubbornAtom Actually they are the same, and you can rewrite it as above. – Daniel Aug 19 '16 at 11:13

Suppose that there are 52 positions in a row where you can put 13 spades into different positions there. So you'll have $\binom{52}{13}$ choices in total. And now you want there to be $a$ spades in the first 13 positions, $b$ spades in the second 13 positions, $c$ spades in the third 13 positions, and $d$ spades in the fourth 13 positions, where $a+b+c+d=13$. So in this case you'll have $\binom{13}{a}\,\binom{13}{b}\,\binom{13}{c}\,\binom{13}{d}$ choices in total. Thus the result follows.