Find the coefficient $x^{8}$ in the product ${(1 + 2 x)}^{6} {(1 - x)}^{7}$ using binomial therm.

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Solution

Given,

$(1 + 2 x)^{6} (1 - x)^{7}$

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

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

$(1 - x)^{7} =^{7} C_{0} -^{7} C_{1} (x) +^{7} C_{2} (x)^{2} -^{7} C_{3} (x)^{3} +^{7} C_{4} (x)^{4} -^{7} C_{5} (x)^{5} +^{7} C_{6} (x)^{6} -^{7} C_{7} (x)^{7}$

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

multiplying (1) and (2), we get,

$(1 + 2 x)^{6} (1 - x)^{7} = (1 + 12 x + 60 x^{2} + 160 x^{3} + 240 x^{4} + 192 x^{5} + 64 x^{6}) (1 - 7 x + 21 x^{2} - 35 x^{3} + 35 x^{4} - 21 x^{5} + 7 x^{6} - x^{7})$

$x^{8}$ coefficients are,

$= (12 x) (- x^{7}) + (60 x^{2}) (7 x^{6}) + (160 x^{3}) (- 21 x^{5}) + (240 x^{4}) (35 x^{4}) + (192 x^{5}) (- 35 x^{3}) + (64 x^{6}) (21 x^{2})$

$= - 12 x^{8} + 420 x^{8} - 3360 x^{8} + 8400 x^{8} - 6720 x^{8} + 1344 x^{8}$

$= 72 x^{8}$

coefficeint of $x^{8} = 72$

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