Binomial Theorem 1 Question 18
19.
Coefficient of $x^{11}$ in the expansion of $\left(1+x^{2}\right)^{4}\left(1+x^{3}\right)^{7}\left(1+x^{4}\right)^{12}$ is
(2014 Adv.)
(a) 1051
(b) 1106
(c) 1113
(d) 1120
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Answer:
Correct Answer: 19. (c)
Solution:
- Coefficient of $x^{r}$ in $(1+x)^{n}$ is ${ }^{n} C_{r}$.
In this type of questions, we find different composition of terms where product will give us $x^{11}$.
Now, consider the following cases for $x^{11}$ in
$\left(1+x^{2}\right)^{4}\left(1+x^{3}\right)^{7}\left(1+x^{4}\right)^{12}$.
Coefficient of $x^{0} x^{3} x^{8}$; Coefficient of $x^{2} x^{9} x^{0}$
Coefficient of $x^{4} x^{3} x^{4}$; Coefficient of $x^{8} x^{3} x^{0}$
$={ }^{4} C_{0} \times{ }^{7} C_{1} \times{ }^{12} C_{2}+{ }^{4} C_{1} \times{ }^{7} C_{3} \times{ }^{12} C_{0}+{ }^{4} C_{2} \times{ }^{7} C_{1}$ $\times{ }^{12} C_{1}+{ }^{4} C_{4} \times{ }^{7} C_{1} \times{ }^{12} C_{0}$
$=462+140+504+7=1113$
