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JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Binomial Theorem

The value of

\sum_{r = 0}^{6} (^{6} C_{r} \cdot^{6} C_{6 - r})

is equal to:

(A)1024
(B)924✓ Correct
(C)1324
(D)1124

Answer: (B)

Solution

Using the identity

(r n) = (n - r n)

, we have

(6 - r 6) = (r 6)

. Thus

\sum_{r = 0}^{6} (r 6) (6 - r 6) = \sum_{r = 0}^{6} (r 6)^{2}

. By Vandermonde's identity,

\sum_{r = 0}^{n} (r n)^{2} = (n 2 n)

. For

n = 6

, this is

(6 12) = 924

.

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Single correct option

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