I've been tasked with a problem in combinatorics. It boils down to this:
We're given a bucket full of balls. There are 52 balls in total, each colored in one of four colors. The colors are evenly distributed. Now, we're tasked to randomly pull out five balls from the bucket.
What's the probability of the selection of five containing balls of all four colors?
How do you go about on solving this?
Quite obviously, you have to find the total amount of possiblities to reach the event of having all four colors in one selection, and divide that by the total amount of possible selections, which is $\binom{52}{5}$.
So, how do we find the total amount of possibilities for this event?
We have to draw two balls of one color and one ball of each of the other three colors. There are $4$ ways to choose the distinguished color, ${13\choose 2}$ ways to choose two balls of that color, and $13$ ways to choose a ball from each of the other colors, so $$4\cdot{13\choose2}\cdot 13^3$$