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Find the coefficient of x¹⁰⁰ in the expansion of [(1 + x)¹⁰⁰ * (1 - x¹⁰¹)] / (1 - x).

1. 1
2. 200
3. C(100,2)
4. 2¹⁰⁰

Correct answer: 2¹⁰⁰

Solution

Since (1-x¹⁰¹)/(1-x) = sumₖ₌₀¹⁰⁰ x^k, the coefficient of x¹⁰⁰ in (1+x)¹⁰⁰ * sumₖ₌₀¹⁰⁰ x^k equals sumₖ₌₀¹⁰⁰ C(100, 100-k) = sum_(j=0)¹⁰⁰ C(100,j) = 2¹⁰⁰.

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