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JEE Main202312 Apr 2023Morning ShiftMathematicsBinomial TheoremActual

If 1 n + 1 n C n + 1 n n C n - 1 + . . . + 1 2 n C 1 + n C 0 = 1023 10 then n is equal to

Options

  1. A9
  2. B8
  3. C7
  4. D6

Correct answer

A. 9

Step-by-step solution

Given that 1 n + 1   n C n + 1 n   n C n - 1 + . . . + 1 2   n C 1 + n C 0 = 1023 10 ⇒ ∑ r = 0 n 1 r + 1   n C r = 1023 10   ∴   n + 1 C r + 1 = n + 1 r + 1   n C r ⇒ ∑ r = 0 n 1 n + 1   n + 1 C r + 1 = 1023 10 ⇒ 1 n + 1 n + 1 C 1 + n + 1 C 2 + . . . + n + 1 C n + 1 = 1023 10 We know that C 0 n + C 1 n + C 2 n + . . . . . . . + C n n = 2 n ⇒ 2 n + 1 - 1 n + 1 = 1023 10 = 2 10 - 1 10 ⇒ n + 1 = 10 ⇒ n = 9 Hence this is the c

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