Indistinguishable objects into distinguishable "bins"

Question

Nehemiah has 10 identical pamphlets which he wishes to put in 5 mail boxes. In how many different ways may he put the pamphlets in such a way that one of the mail boxes has 1 pamphlet while 2 of the mail boxes have exactly 3 pamphlets each? The answer is 120 but I have no idea as to how to get there. Please help

Answer 1

First, we select the mailbox that receives one pamphlet. This is done in $\binom{5}{1} = 5$ ways. Next, we select the two mailboxes that receive $3$ pamphlets. There are $\binom{4}{2} = 6$ such selections. These selections are independent. So by the rule of product, we multiply to obtain $30$ selections.

Now we have $3$ remaining pamphlets to distribute to $2$ mailboxes. It is easy to see there are four such arrangements: $(3, 0), (0, 3), (2, 1), (1, 2)$. Again, by rule of product, there are $30 * 4 = 120$ selections.