JEE Main202623 January 2026Morning ShiftMathematicsBinomial TheoremActual

The value of ¹⁰⁰ C ₅₀ 51 + ¹⁰⁰ C ₅₁ 52 + .+ ¹⁰⁰ C ₁₀₀ 101 is :

Options

A2¹⁰¹ 100
B2¹⁰⁰ 100
C2¹⁰¹ 101
D2¹⁰⁰ 101

Correct answer

D. 2¹⁰⁰ 101

Step-by-step solution

Using n r r+1 = 1 n+1 n+1 r+1 : _ r=50 ¹⁰⁰ 100 r r+1 = 1 101 _ r=50 ¹⁰⁰ 101 r+1 = 1 101 _ k=51 ¹⁰¹ 101 k . Since 101 is odd, by symmetry _ k=51 ¹⁰¹ 101 k = _ k=0 ⁵⁰ 101 k = 2¹⁰¹ 2 = 2¹⁰⁰ . Required value = 2¹⁰⁰ 101 .

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