If the coefficient of x 30 in the expansion of ( 1 + 1 x ) 6 ( 1 + x 2 ) 7 ( 1 - x 3 ) 8 ; ? x ? 0 is, then equals ______ . [2024]

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Solution

Q.
If the coefficient of $x^{30}$ in the expansion of ${(1 + \frac{1}{x})}^{6} {(1 + x^{2})}^{7} {(1 - x^{3})}^{8}; x \neq 0$ is $α,$ then $| α |$ equals ______ . [2024]

Ans.
(678)

We have, ${(1 + \frac{1}{x})}^{6} {(1 + x^{2})}^{7} {(1 - x^{3})}^{8}, x \neq 0$

$= (1 + \frac{6}{x} + \frac{15}{x^{2}} + \frac{20}{x^{3}} + \frac{15}{x^{4}} + \frac{6}{x^{5}} + \frac{1}{x^{6}}) {(1 + x^{2})}^{7} {(1 - x^{3})}^{8}$

$= (1 + \frac{6}{x} + \frac{15}{x^{2}} + \frac{20}{x^{3}} + \frac{15}{x^{4}} + \frac{6}{x^{5}} + \frac{1}{x^{6}}) \times (1 + 7 x^{2} + 21 x^{4} + 35 x^{6} + 35 x^{8} + 21 x^{10} + 7 x^{12} + x^{14})$

$1 - 8 x^{3} + 28 x^{6} - 56 x^{9} + 70 x^{12} - 56 x^{15} + 28 x^{18} - 8 x^{21} + x^{24}$

Coefficient of $x^{30}$ in the expansion of ${(1 + \frac{1}{x})}^{6} {(1 + x^{2})}^{7} {(1 - x^{3})}^{8}$

$= 28 \times 7 + 28 \times 15 - 8 \times 21 \times 6 - 8 \times 20 \times 7 - 8 \times 6 + 15 \times 35 + 35 + 15 \times 21 + 7$

$= 196 + 420 - 1008 - 1120 - 48 + 525 + 35 + 315 + 7$

$= - 678$

$∴$ $| α | = 678$

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