First slide

Binomial theorem for positive integral Index

Question

The coefficient of x⁵ in the expansion of $(1 + 2 x)^{6} (1 - x)^{7}$ is

Moderate

A

192
B

171
C

21
D

None of these

Solution

$(1 + 2 x)^{6} (1 - x)^{7} = [1 +^{6} C_{1} (2 x) +^{6} C_{2} (2 x)^{2} +^{6} C_{3} (2 x)^{3} +^{6} C_{4} (2 x)^{4} +^{6} C_{5} (2 x)^{5} +^{6} C_{6} (2 x)^{6}] \times [1 -^{7} C_{1} x +^{7} C_{2} x^{2} -^{7} C_{3} x^{3} +^{7} C_{4} x^{4} \dots]$
$= (1 + 12 x + 60 x^{2} + 160 x^{3} + 240 x^{4} + 192 x^{5} + \dots) \times (1 - 7 x + 21 x^{2} - 35 x^{3} + 35 x^{4} - 21 x^{5} + \dots)$
$∴ Coefficient of x^{5} in the expansion of (1 + 2 x)^{6} (1 - x)^{7}$
$= 1 \times (- 21) + 12 \times 35 + 60 \times (- 35) + 160 \times 21 + 240 \times (- 7) + 192$
$= - 21 + 420 - 2100 + 3360 - 1680 + 192$
=171

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