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Question

Coefficient of x9 in (1+x)((1+x2)(1+x3)......(1+x100))?

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Solution

Coefficient of x9
(1+x)(1+x2)(1+x3)..(1+x100)
First observe that x9 can be only generate by products of the form.
1.x9,x2.x7,x4.x3.x2, x4.x3.x2 and so on.
Therefore all we need to find is the number of ways the integer g can be partitioned into integers from the set.
{1,2,3,4,5,6,7,8,9} These are
{9},{1.8}, {1,2,6},{1,3,5},{2,7},{3,6},{4,5}
{2,3,4}
There are 8 such terms. Therefore the coefficient of x9 is 8.

1191444_1157999_ans_b89fa61a361a4fe3a334c3aea1eb84b1.jpg

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