Coefficient of expansion

Probability Level 3

For what integer n n ( 0 ≤ n ≤ 9 ) (0\le n\le 9) are the coefficients of x x and x 2 {x}^{2} both negative in the expansion of ( 1 − x ) n ( 1 + x ) 9 − n { (1-x) }^{ n }{ (1+x) }^{ 9-n } ?


The answer is 5.

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Lawahar Qasid by Lawahar Qasid

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1 solution

Aaaaa Bbbbb
Jul 13, 2014

When n=5, the expression is: ( 1 − 5 x + 10 x 2 + . . . − x 5 ) ( 1 + 4 x + 6 x 2 + 4 x 3 + x 4 ) = 1 − x − 4 x 2 + . . . (1-5x+10x^2+...-x^5)(1+4x+6x^2+4x^3+x^4)=1-x-4x^2+... ⇒ n = 5 \Rightarrow n=\boxed{5}

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