What are the odds of rolling 3 pairs with six dice?

Question

Given a roll of six standard six-side dice, how do you calculate the odds of rolling 3 pairs? This should included non-unique pairs like 2,2,2,2,3,3 or even 5,5,5,5,5,5.

Answer 1

Let's split into $3$ cases:

Case 1: All dice show the same number. There are $6$ ways this can happen.

Case 2: $4$ dice show the same number, and $2$ dice show a different number. There are $\binom{6}{2}=15$ different ways to choose the $2$ dice that show a different number, and $\binom{6}{2}=15$ ways to choose which two numbers appear. We then multiply by $2$ to choose which number appears $4$ times, and which number appears $2$ times. Thus there are $15\cdot 15\cdot 2=450$ possible outcomes for Case 2.

Case 3: There are three distinct pairs. Here, there $\binom{6}{3}=20$ ways to choose which three numbers appear, and $6!/2^3=90$ ways to arrange them. So here we have $1800$ possibilities.

Altogether, there are $6^6$ different outcomes, which gives us a probability of:

$$\frac{6+450+1800}{6^6}=\frac{47}{972}\approx.04835$$

It looks like this will happen about once every $21$ throws.