Combinatorial Sum

Probability Level 4

r=1nr3(nr)=2n3(n3+an2+bn+c)\sum_{r=1}^{n}r^3 {n \choose r} = 2^{n-3}\left(n^3+an^2+bn+c\right)

Find the value of a2+b2+c2a^2+b^2+c^2 if the above equation is true for all integers rr and nn.

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