Question

A person rolls a standard six-sided die 8 times. In how many ways can he get 3 fives, 4 sixes, and 1 two?

Answer 1

With a standard binomial, it becomes a fraction where the numerator is the factorial of the total, the 2 items in the denominator are the factorials of the 2 things that combine to the total, the successes and the failures.

In this case, we have a total number of items, but there are 3 groups that combine to the total, this works very similar to the binomial. The numerator is still the factorial of the total, but in this case we will have 3 items in the denominator to represent the 3 different rolls.

8! / 3! 4! 1! = 280 ways

This can be expanded to more than 3 states the same way. This is how the pmf of the multinomial probability distribution begins.