2015/16 AHSMC Part 1 Problem 13

$$(1 + x + x^2 + \cdots + x^{50})^3 = (1 + x + x^2 + \cdots + x^{50}) \cdot (1 + x + x^2 + \cdots + x^{50}) \cdot (1 + x + x^2 + \cdots + x^{50}).$$ The coefficient of $x^{50}$ is just the number of $x^a x^b x^c = x^{a+b+c}$ with $a+b+c = 50$, $a,b,c \in \{0,1,\ldots,50\}$. If $a = 0$ the equation $b+c = 50$ has $51$ solutions, namely $(0,50)$, $(1,49)$, $\ldots$, $(50,0)$. Also, if $a = 1$, the equation $b+c = 49$ has $50$ solutions and so on. The number of all solutions is $$51 + 50 + \cdots + 1 = 1326.$$ Hence $c_{50} = 1326$. The answer is (C).