If $C_0, C_1, C_2, \dots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$ then the value of $\sum r^3 \cdot C_r$ when $n = 5$ is
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For summations involving binomial coefficients, memorize the standard results:
$\sum r C_r = n 2^{n-1}$
$\sum r^2 C_r = n(n+1) 2^{n-2}$
$\sum r^3 C_r = n^2(n+3) 2^{n-3}$
These are very useful for competitive exams.