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JEE Main202324 Jan 2023Morning ShiftMathematicsBinomial TheoremActual

The value of ∑ r = 0 22 C r 22 · C r 23 is

Options

  1. AC 23 45
  2. BC 23 44
  3. CC 24 45
  4. DC 22 44

Correct answer

A. C 23 45

Step-by-step solution

Let, S = ∑ r = 0 22 C r 22 · C r 23 Now using C r n = C n - r n we get, S = ∑ r = 0 22 C r 22 · C 23 - r 23 ⇒ S = C 0 22 · C 23 23 + C 1 22 · C 22 23 + . . . . . . . . . . . . C 22 22 · C 1 23 Which is sum of coefficient of x 23 in the expansion of 1 + x 22 · 1 + x 23 or 1 + x 23 + 22 = 1 + x 45 Hence, ∑ r = 0 22 C r 22 · C 23 - r 23 = C 23 45

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