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

9 can be expressed as a sum of distinct integers in the following ways:

9 = 9 (1 way)
9 = 1 + 8 (1 way)
9 = 2 + 7 (1 way)
9 = 3 + 6 (1 way)
9 = 4 + 5 (1 way)
9 = 1 + 2 + 6 (1 way)
9 = 1 + 3 + 5 (1 way)
9 = 1 + 4 + 4 (This is not valid since integers must be distinct)
9 = 2 + 3 + 4 (1 way)

Thus, there are 5 ways to express 9 as a sum of distinct integers, each less than or equal to 9. There are 3 additional ways when allowing sums of three integers.

Therefore, the coefficient of x⁹ in the given expression is 8.