Counting, Combinations Stuck here :( What's the number of ways a person can score 60 total points from taking 4 tests given that 3 of 4 tests each have maximum score of 50 points and other test a maximum score of 100 points?
My strategy is to enumerate all ways in which scores from each exam sum up to 60 points. Is there some efficient way to do this? 

This can be viewed as finding the number of non-negative integer solutions of the equation $x_1+x_2+x_3+x_4=60$, with the proviso that $x_1$,$x_2$, and $x_3$ are $\le 50$.
We solve the problem using the Stars and Bars procedure. I prefer to think of it as distributing $60$ identical candies between $4$ kids, with the proviso that none of the first $3$ kids gets more than $50$ candies.
With no such restriction, the number of ways is $\binom{60+4-1}{4-1}$.
From this we must subtract the bad ways, where one of the first three kids gets $51$ or more candies. How many ways if the first kid gets $51$ or more? Give her $51$ candies to start with.Then distribute the remaining $9$ among the $4$ kids (so the first kid may get additional candies).  This can be done in $\binom{9+4-1}{3}$ ways. So the total number of "bad" ways is $3\binom{12}{3}$.  It follows that the number of ways to distribute the marks between the tests is $\binom{63}{3}-\binom{12}{3}$.
