Binomial Theorem 1 Question 35
37. The coefficient of $x^{9}$ in the expansion of $(1+x)\left(1+x^{2}\right)\left(1+x^{3}\right) \ldots\left(1+x^{100}\right)$ is
(2015 Adv.)
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Solution:
- Coefficient of $x^{9}$ in the expansion of $(1+x)\left(1+x^{2}\right)\left(1+x^{3}\right) \ldots\left(1+x^{100}\right)=$ Terms having $x^{9}$ $=\left[1^{99} \cdot x^{9}, 1^{98} \cdot x \cdot x^{8}, 1^{98} \cdot x^{2} \cdot x^{7}, 1^{98} \cdot x^{3} \cdot x^{6}\right.$,
$$ \left.1^{98} \cdot x^{4} \cdot x^{5}, 1^{97} \cdot x \cdot x^{2} \cdot x^{6}, 1^{97} \cdot x \cdot x^{3} \cdot x^{5}, 1^{97} \cdot x^{2} \cdot x^{3} \cdot x^{4}\right] $$
$\therefore$ Coefficient of $x^{9}=8$
