Permutations, Combinations, and Counting A group of 63 people are camping together. They have two 6-person tents, three 4-person tents, five 3-person tents, and three 2 person tents. 18 people will sleep outside of the tents under a tarp. 
Determine the how many ways the people can choose the tents to sleep. (All the tents are different)
Suppose exactly 4 people snore. Count the number of ways of assigning the people to the tents so that all the snorers share their tents only with other snorers
 A: The calculation mentioned in the comment began well. Note that the numbers mentioned for the various tents and the tarp add up to $63$, so the tents will be all full.
The people for the tarp can be chosen in $\binom{63}{18}$ ways.
For each such way, the $6$ people in big tent Number 1 can be chosen in $\binom{45}{6}$ ways.
For each such way, the $6$ people in big tent Number 2 can be chosen in $\binom{39}{6}$ ways.
Continue.
When we multiply,  and simplify the binomial coefficients, we get something that begins like this:
$$\frac{63!}{18!45!}\frac{45!}{6!39!}\frac{39!}{6!33!}\frac{33!}{4!29!}\cdots.$$
There is massive cancellation, and we end up with
$$\frac{63!}{18!6!6!4!4!4!3!3!3!3!3!2!2!2!}.$$
You may recognize this as a multinomial coefficient.
Another way of thinking about the problem is that we will line up the people in order of Student Number, or height. Then the assignment to tarp/tents will be made by writing down a $63$-letter word, using $18$ copies of the letter $T$ (tarp), $6$ copies of the letter $A_1$ (big tent Number 1), $6$ copies of the letter $A_2$ (big tent Number 2), and so on. 
For the snoring restriction, the snorers will be (i) in one of the $4$-person tents ($3$ choices) or (ii) in two of the $2$-person tents. For Case (ii), the tents can be chosen in $\binom{3}{2}$ ways, and once the tents have been chosen, the people can be assigned to the two tents in $\binom{4}{2}$ ways.
For each of Cases (i) and (ii), we can calculate the number of ways the rest of the people can be assigned by using the method of the first question. Now put the pieces together. 
Remark: The problem becomes more complicated if we assume that tents of the same size are indistinguishable. 
