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

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Binomial Theorem

The coefficient of

x^{9}

in the expansion of

(1 + x) (1 + x^{2}) (1 + x^{3}) \dots (1 + x^{100})

Answer: 8

Solution

The coefficient of

x^{9}

counts the number of ways to partition 9 using distinct positive integers ≤ 100. The partitions are:

9

(1 way),

1 + 8

2 + 7

3 + 6

4 + 5

(4 ways with 2 parts), and

1 + 2 + 6

1 + 3 + 5

2 + 3 + 4

(3 ways with 3 parts). Total:

1 + 4 + 3 = 8

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Integer answer

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