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Which of the following summation identities are correct? (A) sum_(r=0)³⁰ (-1)^r * C(30,r) * C(90-r, 60) = C(60,30) (B) sum_(r=0)³⁰ (-1)^r * C(30,r) * C(90-r, 30) = 0 (C) sum_(r=0)³⁰ (-1)^r * C(30,r) * C(90-2r, 30) = 2³⁰ (D) sum_(r=0)³⁰ (-1)^r * C(30,r) * C(90-3r, 30) = 1
- A and B only
- B and C only
- A and C only
- A and D only
Correct answer: A and C only
Solution
Each sum is evaluated via the identity: S = coeff of x^p in (1+x)^m * (x/(1+x))ⁿ when k=1. (A) gives C(60,30) — TRUE. (B) gives 1 not 0 — FALSE. (C) with k=2 gives ((1+x)³-1)³⁰ coefficient = 2³⁰ — TRUE. (D) with k=3 gives 3³⁰ not 1 — FALSE.
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