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Question
Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.
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Solution
Given expression is (1 + x + x2 + x3)11
= [(1 + x) + x2 (1 + x)]11
= [(1 + x)(1 + x2)]11
= (1 + x)11 · (1 + x2)11
Expanding the above expression, we get
(11C0 + 11C1x + 11C2x2 + 11C3x3 + 11C4x4 + …) · (11C0 + 11C1x2 + 11C2x4 +)
= (1 + 11x + 55x2 + 165x3 + 330x4 …) · (1 + 11x2 + 55x4 + …)
Collecting the terms containing x4, we get
(55 + 605 + 330)x4 = 990x4
Hence, the coefficient of x4 = 990
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