The number of rolls of $6$ dice with exactly $4$ distinct values.

Question

I know the answer is $${6 \choose 4}{5 \choose 3}$$ However, I don't understand why this is true?

Answer 1

The binomial coefficient $\binom{n}{k}$ is exactly the number of possible choices of $k$ elements in a set with $n$ elements.

The rolls of $6$ dice with exactly $4$ distinct values are constructed as follows:

• Choose the $4$ distinct values among the $6$ possible, this gives you a factor $\binom{6}{4}$.
• Partition your set of dice into $4$ nonempty subsets (one for each value). Since the dice are undistinguishable, this is equivalent to put $3$ separations between elements, choosing between $5$ possible places to put such a separation gives you a factor $\binom{5}{3}$.

An example of such a separation cited above (stars represent dice): $$**|*|**|*$$