Coefficient of x 11 in the expansion of ( 1 + x 2 ) 4 ( 1 + x 3 ) 7 ( 1 + x 4 ) 12 is [2014]

Question

Coefficient of $x^{11}$ in the expansion of ${(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$ is [2014]

Smart Achievers · Accepted Answer

(3)

$Coeff. of x^{11} in expansion of {(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$

$= [Coeff. of x^{a} in {(1 + x^{2})}^{4}] \times [Coeff. of x^{b} in {(1 + x^{3})}^{7}] \times [Coeff. of x^{c} in {(1 + x^{4})}^{12}]$

$Such that a + b + c = 11$

$Here a = 2 m, b = 3 n, c = 4 p$

$∴$ $2 m + 3 n + 4 p = 11$

$Case I: m = 0, n = 1, p = 2$

$Case II: m = 1, n = 3, p = 0$

$Case III: m = 2, n = 1, p = 1$

$Case IV: m = 4, n = 1, p = 0$

$∴$ $Required coefficient$

$= {}^{4}C_{0} \times {}^{7}C_{1} \times {}^{12}C_{2} + {}^{4}C_{1} \times {}^{7}C_{3} \times {}^{12}C_{0} + {}^{4}C_{2} \times {}^{7}C_{1} \times {}^{12}C_{1} + {}^{4}C_{4} \times {}^{7}C_{1} \times {}^{12}C_{0}$

$= 462 + 140 + 504 + 7 = 1113$

Solution

Q.
Coefficient of $x^{11}$ in the expansion of ${(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$ is [2014]

1 1051

2 1106

3 1113

4 1120

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