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JEE Main202325 Jan 2023Evening ShiftMathematicsBinomial TheoremActual

∑ k = 0 6 C 3 51 - k is equal to

Options

  1. AC 4 51 - C 4 45
  2. BC 3 51 - C 3 45
  3. CC 4 52 - C 4 45
  4. DC 3 52 - C 3 45

Correct answer

C. C 4 52 - C 4 45

Step-by-step solution

Given, ∑ r = 0 6 C 3 51 - k = C 3 51 + C 3 50 + C 3 49 + C 3 48 + C 3 47 + C 3 46 + C 3 45 Now we know that C r n + C r + 1 n = C r + 1 n + 1 or C r n = C r + 1 n + 1 - C r + 1 n Now using the above formula in given expression we get, ∑ r = 0 6 C 3 51 - k = C 3 51 + C 3 50 + C 3 49 + C 3 48 + C 3 47 + C 3 46 + C 4 46 ⏟ - C 4 45 ⇒ ∑ r = 0 6 C 3 51 - k = C 3 51 + C 3 50 + C 3 49 + C 3 48 + C 3 47 + C 4 47 ⏟ - C 4 45 ⇒ ∑ r = 0 6 C 3 51 - k = C 3 51 + C 3 50 + C 3 49 + C

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