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Question

Find the coefficient of x4 in the expansion of (1+x+x2+x3)11.

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Solution

(1+x+x2+x3)11
((1+x)(1+x2))11
(1+x)11(1+x2)11
[11C0(x0)+11C1(x1)+11C2(x2)+.....+11C11(x11)][11C0(x2)0+11C1(x2)1+11C2(x2)2+.....+11C11(x2)11]
Coefficient of x4 is
11C2.11C0+11C1.11C2+11C0.11C4
=11!9!2!.1+11!10!1!.11!9!2!+11!4!7!.1
=11×5+11×11×5+110×3=55+605+330=990

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