Find the coefficient of $x^{5}$ in the product $(1 + 2 x)^{6} (1 - x)^{7}$ using binomial theorem.

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Solution

Coefficient of $x^{5}$ in the product of $(1 + 2 x)^{6} (1 - x)^{7}$
$(1 + 2 x)^{6} (1 - x)^{7}$
Expanding using Binomial theorem
$1 +^{6} C_{1} (2 x) +^{6} C_{2} (2 x)^{2} +^{6} C_{3} (2 x)^{3} +^{6} C_{4} (2 x)^{4} +^{6} C_{5} (2 x)^{5} +^{6} C_{6} (2 x)^{6}] \times [1 -^{7} C_{1} x +^{7} C_{2} x^{2} -^{7} C_{3} x^{3} +^{7} C_{4} x^{4} -^{7} C_{5} x^{5} . . .]$
$= [1 + 12 x + 60 x^{2} + 160 x^{2} + 240 x^{4} + 192 x^{5} + 64 x^{6}] \times [1 - 7 x + 21 x^{2} - 35 x^{3} + 35 x^{4} - 21 x^{5} . . . .]$
Now, the coefficient of $x^{5}$ in the product is $= 1 \times (- 21) + 12 \times 35 + 60 \times (- 35) + 160 \times 21 + 240 \times (- 7) + 192 \times 1$
$= - 21 + 420 - 2100 + 3360 - 1680 + 192$
$= 171$
Coefficient of $x^{5}$ is $171$ .

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