First slide

Binomial theorem for positive integral Index

Question

Coefficient of x¹¹ in the expansion of ${(1 + x^{2})}^{4} {(1 + x^{3})}^{7}$ ${(1 + x^{4})}^{12}$ is

Moderate

A

1051
B

1106
C

1113
D

1120

Solution

$2 x_{1} + 3 x_{2} + 4 x_{3} = 11$
Possibilities are $(0, 1, 2); (1, 3, 0); (2, 1, 1); (4, 1, 0)$
$∴ Required coefficients$
$= (^{4} C_{0} \times^{7} C_{1} \times^{12} C_{2}) + (^{4} C_{1} \times^{7} C_{3} \times^{12} C_{0}) + (^{4} C_{2} \times^{7} C_{1} \times^{12} C_{1}) + (^{4} C_{4} \times^{7} C_{1} \times 1) \begin{matrix} = (1 \times 7 \times 66) + (4 \times 35 \times 1) + (6 \times 7 \times 12) + (1 \times 7) \\ = 462 + 140 + 504 + 7 = 1113 \end{matrix}$

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