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JEE Main 2023 · 25 Jan · Shift 2

JEE Main 2023 Mathematics question on Binomial Theorem

∑k=0651−kC3\sum_{k=0}^{6} 51-k C_3 is equal to
  1. A.
    51C3−45C3^{51}C_3 - ^{45}C_3
  2. B.
    51C4−45C4^{51}C_4 - ^{45}C_4
  3. C.
    52C4−45C4^{52}C_4 - ^{45}C_4
  4. D.
    52C3−45C3^{52}C_3 - ^{45}C_3
Show answer and solution

Correct option: C

Solution

Using the Hockey Stick identity: Σ(r=0 to n) C(a-r, k) = C(a+1, k+1) - C(a-n, k+1).

Here the sum is C(51,3) + C(50,3) + C(49,3) + C(48,3) + C(47,3) + C(46,3) + C(45,3), which is 7 terms (k=0 to 6).

By the Hockey Stick identity: C(45,3) + C(46,3) + ... + C(51,3) = C(52,4) - C(45,4).

This is because Σ(r=r₀ to n) C(r,m) = C(n+1, m+1) - C(r₀, m+1), so Σ(r=45 to 51) C(r,3) = C(52,4) - C(45,4).

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