# Combinatorics homework question

The answer is 54912.

This is what I've tried so far: So first you have to pick a rank to occur 3 times so thats 13, now you gotta pick a suit that that rank has, which is now 13 * 4. Now you need to pick that same rank 2 more times. The second time it will be 13 * 3, then the third time it will be 13 * 2. So the expression so far is (13^3)*(4*3*2) but that number is already too high when I try to add the other two cards. What am I doing wrong?

The key to this question is the terms 'combination' and 'subset'. You want to model this problem with combinations for each subset/property and then multiply these together.

That is: rank $\times$ suit $\times$ other_rank $\times$ other_suits.

There are 13 ranks so we choose one with $\binom{13}{1}$, choose our suits with $\binom{4}{3}$, chose our other two ranks $\binom{12}{2}$ and then the suits for these two $\binom{4}{1}^2$.
So the answer is $\binom{13}{1}\binom{4}{3}\binom{12}{2}\binom{4}{1}^2=54912$.
$13 \cdot {4 \choose 3} \cdot {12 \choose 2} \cdot 4 \cdot 4 = 54912$.