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

Correct answer: C. 1113 2 x _ 1 + 3 x _ 2 + 4 x _ 3 = 11 Possibilities are (0, 1, 2); (1, 3, 0); (2, 1, 1); (4, 1, 0) ∴ Required coefficients = ( ^ 4 C _ 0 × ^ 7 C _ 1 × ^ 12 C _ 2 ) + ( ^ 4 C _ 1 × ^ 7 C _ 3 × ^ 12 C _ 0 ) + ( ^ 4 C _ 2 × ^ 7 C _ 1 × ^ 12 C _ 1 ) + ( ^ 4 C _ 4 × ^ 7 C _ 1 × 1 ) = (1 × 7 × 66) + (4 × 35 × 1) + (6 × 7 × 12) + (1 × 7) = 462 + 140 + 504 + 7 = 1113

JEE Advanced 2014Paper 2

Exam: JEE Advanced
Year: 2014
Paper: Paper 2
Subject: Mathematics
Chapter: Binomial Theorem

JEE Advanced 2014 · Paper 2 · Mathematics · Binomial Theorem

Coefficient of x ^ 11 in the expansion of (1 + x ^ 2 ) ^ 4 (1 + x ^...

Coefficient of

{x}^{11}

in the expansion of

{\left(1 + {x}^{2}\right)}

^{4}{

\left(1 + {x}^{3}\right)}^{7}{\left(1 + {x}^{4}\right)}^{12}

Options

A1051
B1106
C1113
D1120

Correct answer

C. 1113

Step-by-step solution

2{x}_{1} + 3{x}_{2} + 4{x}_{3} = 11

<br >Possibilities are \left(0, 1, 2\right);

\left(1, 3, 0\right); \left(2, 1, 1\right); \left(4, 1, 0\right)

∴

Required coefficients

= \left({}^{4}{C}_{0} \times {}^{7}{C}_{1} \times {}^{12}{C}_{2}\right)

+ \left({}^{4}{C}_{1} \times {}^{7}{C}_{3} \times {}^{12}{C}_{0}\right)

+ \left({}^{4}{C}_{2} \times {}^{7}{C}_{1} \times {}^{12}{C}_{1}\right) + ( {}^{4}{C}_{4} \times {}^{7}{C}_{1} \times 1 )

= \left(1 \times 7 \times 66\right) + \left(4 \times 35 \times 1\right) + \left(6 \times 7 \times 12\right) + \left(1 \times 7\right) <br >

= 462 + 140 + 504 + 7 = 1113

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