The coefficient of $$x9$$ in the expansion of (1$$ $$+$$ $$x) (1$$ $$+$$ $$x2) (1$$ $$+$$ $$x3) ... (1$$ $$+$$ $$x100) is $$_______$$.

Binomial theorem for positive integral Index

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Question

The coefficient of x⁹ in the expansion of (1 + x) (1 + x²) (1 + x³) ... (1 + x¹⁰⁰) is _______.

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Solution

x⁹ can be formed in following eight ways.
i.e., $x^{9}, x^{1 + 8}, x^{2 + 7}, x^{3 + 6}, x^{4 + 5}, x^{1 + 2 + 6}, x^{1 + 3 + 5}, x^{2 + 3 + 4}$
and coefficient in each case is 1.
$∴$ Coefficient of x⁹ = 1 + 1 + 1 +... 8 times
= 8

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