<s>[INST] In how many ways can a student score exactly $100$ points on four $50$ point exams? greater than $100$? A student takes up $4$ exams with $50$ points each. In how many ways can he score exactly $100$? Similarly, in how many ways can he score greater than $100$?
My try:
I cannot seem to fill in the $4$ possibilities. The range depends on the constraint of equating to $100$.
Thanks for any help!
 [/INST] The scores he can get  are the coefficients of the different powers of $x$ in the expansion
\begin{align*}
(1+x+\cdots + x^{50})^4 &= \left(\frac{1-x^{51}}{1-x}\right)^4 \\
&= (1-4x^{51} + 6X^{102} - 4X^{153} + X^{204})\left(1+ 4x + \frac{4\cdot 5}{1\cdot 2}x^2 + \frac{4\cdot 5 \cdot 6}{3!}x^3 + \cdots +\binom{n+3}{3}x^n+\cdots \right)
\end{align*}
Hence the number of ways in which he can get exactly 100 is 
$$ \binom{103}{3} - 4 \binom{52}{3} = 88451$$
Second part has already been solved in the other solutions posted.
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