Question: The formula for the number of multisets is (n + k - 1)! divided by a product of two other factorials. We seek an explanation using the quotient principle of why this counts multisets. The formula for the number of multisets is also a binomial coefficient, so it should have an interpretation involving choosing k items from n + k - 1 items. The parts of the problem that follow lead us to these explanations.
(a) In how many ways may we place k red checkers and n - 1 black checkers in a row?
(b) How can we relate the number of ways of placing k red checkers and n - 1 black checkers in a row to the number of k-element multisets of an n-element set, say the set {1, 2,...,n} to be specific?
(c) How can we relate the choice of k items out of n + k - 1 items to the placement of red and black checkers as in the previous parts of this problem?