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

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Solution

By using the formula :
$(a + b)^{n} = n_{C_{0}} a^{n} + n_{C_{1}} a^{n - 1} b + n_{C_{2}} a^{n - 2} b^{2} + . . . . . . . . . n_{C_{r}} a^{n - r} b^{r} + . . . . . . n_{C_{n}} b^{n}$

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

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

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

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

$= 1 - 7 x + 21 x^{2} - 35 x^{3} + 35 x^{4} - 21 x^{5} + 7 x^{6} - x^{7}$ -----------(ii)
multiply equation (i) and (ii) .

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

$= - 21 + 420 - 2100 + 3360 - 1680 + 192$
$= 171$

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