In the product of expression of $(1 + 2 x)^{6} (1 - x)^{7}$ , find the coefficient of $x^{5}$ ?

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Solution

${(1 + 2 x)}^{6} = 6 C_{0} 1 + 6 C_{1} (2 x) + 6 C_{2} {(2 x)}^{2} + 6 C_{2} {(2 x)}^{3} + 6 C_{4} {(2 x)}^{4} + 6 C_{5} {(2 x)}^{5} + 6 C_{6} {(2 x)}^{6}$

$= 1 + 12 x + 60 x^{2} + 20 \times 8 x^{3} + 15 \times 16 x^{4} + 6 \times 32 x^{6} + 64 x^{5}$

$= 1 + 12 x + 60 x^{2} + 160 x^{3} + 240 x^{4} + 192 x^{5} + 64 x^{6}$

${(1 - x)}^{7} = 1 - 7 C_{1} x + C_{2} x^{2} - C_{3} x^{3} + 7 C_{4} x^{4} - 7 C_{5} x^{5} + 7 C_{6} x^{6} - 7 C_{7} x^{7}$

$1 - 7 x + {12 x}^{2} - 35 x^{3} + 35 x^{4} - 21 x^{5} + 7 x^{6} - x^{7} ⟹ (2)$

Coefficient of $x^{5}$ in the product $(1 + 2 x)^{6} {(1 - x)}^{7}$

$= 1 \times - 21 + 12 \times 35 + 60 \times - 35 + 160 \times 21 + 240 \times - 7 + 192 \times 1$

$= - 21 + 420 - 2100 + 3360 - 1680 + 192$

$= 171$

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