HMMT-Nov 2009 guts, Problem 19

You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak point until you have cut off all of its claws and legs. In how many ways can you defeat the giant enemy crab? (Note that the legs are distinguishable, as are the claws.)

Solution: The answer is given by $6 ! 2 !\left(\begin{array}{l}5 \\ 2\end{array}\right)$, because we can cut off the claws and legs in any order and there are $\left(\begin{array}{l}5 \\ 2\end{array}\right)$ ways to decide when to cut off the two claws (since we can do it at any time among the last 5 cuts). $\fbox{14400}$.

Answer: 14400