First slide

Binomial theorem for positive integral Index

Question

The coefficient of x⁵ in the expansion of ${(x^{2} - x - 2)}^{5}$ is

Moderate

A

-83
B

-82
C

-86
D

-81

Solution

$\begin{array}{l} (x - 2)^{5} (x + 1)^{5} \\ = [^{5} C_{0} x^{5} -^{5} C_{1} x^{4} \times 2 +^{5} C_{2} x^{3} \times 2^{2} -^{5} C_{3} x^{2} \times 2^{3} +^{5} C_{4} x \times 2^{4} -^{5} C_{5} \times 2^{5}] [^{5} C_{0} +^{5} C_{1} x +^{5} C_{2} x^{2} +^{5} C_{3} x^{3} +^{5} C_{4} x^{4} +^{5} C_{5} x^{5}] \end{array}$
$\Rightarrow$ Coefficient of x⁵
$=^{5} {C_{0}}^{5} C_{5} -^{5} C_{1} \times 2 \times^{5} C_{4} +^{5} C_{2} \times 2^{2} \times^{5} C_{3} -^{5} C_{3} \times 2^{3} \times^{5} C_{2} +^{5} C_{4} \times 2^{4} \times^{5} C_{1} -^{5} C_{5} \times 2^{5} \times^{5} C_{0} = 1 - 5 \times 5 \times 2 + 10 \times 10 \times 4 - 10 \times 10 \times 8 + 5 \times 5 \times 16 - 32 = - 81$

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