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Question: The coefficient of \(x^{5}\) in the expansion of \((x^{2} - x - 2)^{5}\) is...

The coefficient of x5x^{5} in the expansion of (x2x2)5(x^{2} - x - 2)^{5} is

A

– 83

B

– 82

C

– 81

D

0

Answer

– 81

Explanation

Solution

Coefficient of x5x^{5} in the expansion of (x2x2)5(x^{2} - x - 2)^{5} is

5!n1!.n2!n3!(1)n1(1)n2(2)n3\sum_{}^{}\frac{5!}{n_{1}!.n_{2}!n_{3}!}(1)^{n_{1}}( - 1)^{n_{2}}( - 2)^{n_{3}}.

where n1+n2+n3=5n_{1} + n_{2} + n_{3} = 5 and n2+2n3=5n_{2} + 2n_{3} = 5. The possible value of n1,n2n_{1},n_{2} and n3n_{3} are shown in margin

n1n_{1} n2n_{2} n3n_{3}

1 3 1

2 1 2

0 5 0

\therefore The coefficient of x5x^{5}

= 5!1!3!1!(1)1(1)3(2)1+5!2!1!2!(1)2(1)1(2)2\frac{5!}{1!3!1!}(1)^{1}( - 1)^{3}( - 2)^{1} + \frac{5!}{2!1!2!}(1)^{2}( - 1)^{1}( - 2)^{2} +

5!0!5!0!(1)0(1)5(2)0\frac{5!}{0!5!0!}(1)^{0}( - 1)^{5}( - 2)^{0} = 401201=8140 - 120 - 1 = - 81