Let C r n denote the binomial coefficient of x r in the expansion of ( 1 + x ) n . If ∑ k = 0 10 2 2 + 3 k C k n = α . 3 10 + β · 2 10 , α , β ∈ R , then α + β is equal to ___ .

Question

Let C r n denote the binomial coefficient of x r in the expansion of ( 1 + x ) n . If ∑ k = 0 10 2 2 + 3 k C k n = α . 3 10 + β · 2 10 , α , β ∈ R , then α + β is equal to ___ .,

MARKS App · Accepted Answer

Instead of C k n it must be C k 10 i.e. ∑ k = 0 10 2 2 + 3 k C k 10 = α · 3 10 + β · 2 10 L H S = 4 ∑ k = 0 10 C k 10 + 3 ∑ k = 0 10 k · 10 k · C k - 1 9 = 4 . 2 10 + 3 . 10 . 2 9 = 19 . 2 10 = α . 3 10 + β . 2 10 ⇒ α = 0 , β = 19 ⇒ α + β = 19

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