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

JEE AdvancedJEE Advanced 2014Binomial Theorem

Solution:

$2 x _{1}+3 x _{2}+4 x _{3}=11$
Possibilities are $(0,1,2) ;(1,3,0) ;(2,1,1) ;(4,1,0)$.
$\therefore $ 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)+\left({ }^{4} C _{4} \times{ }^{7} C _{1} \times 1\right) $
$=(1 \times 7 \times 66)+(4 \times 35 \times 1)+(6 \times 7 \times 12)+(1 \times 7) $
$=462+140+504+7=1113$