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The coefficient of x^5  in the expansion of \left(x^2-x-2\right)^5 \text { is }

Option: 1

– 83


Option: 2

– 82


Option: 3

– 81


Option: 4

0

 


Answers (1)

best_answer

Coefficient of  x^5  in the expansion of \left(x^2-x-2\right)^5 \text { is } \sum \frac{5 !}{n_{1} ! n_{2} ! n_{3} !}(1)^{n_1}(-1)^{n_2}(-2)^{n_3}

wheren_1+n_2+n_3=5 \text { and } n_2+2 n_3=5  The possible value ofn_1, n_2 \text { and } n_3 are shown in margin 

\begin{array}{lll} n_1 & n_2 & n_3 \\ 1 & 3 & 1 \\ 2 & 1 & 2 \\ 0 & 5 & 0 \end{array}

The coefficient of x^5=\frac{5 !}{1 ! 3 ! 1 !}(1)^1(-1)^3(-2)^1+\frac{5 !}{2 ! 1 ! 2 !}(1)^2(-1)^1(-2)^2+\frac{5 !}{0 ! 5 ! 0 !}(1)^0(-1)^5(-2)^0=

40-120-1=-81

 

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