First slide

Binomial theorem for positive integral Index

Question

The value of $\sum_{r = 0}^{6} (^{6} C_{r} .^{6} C_{6 - r})$ is equal to:

Moderate

A

924
B

1324
C

1024
D

1124

Solution

The given expression is $\sum_{r = 0}^{6} 6 C_{r} 6 C_{6 - r} = 6 C_{0} .6 C_{6} + 6 C_{1} .6 C_{5} + ....... + 6 C_{6} .6 C_{0}$
$\begin{array}{l} We know that (1 + x)^{6} (1 + x)^{6} = [6 C_{0} + 6 C_{1} x + \dots \dots + 6 C_{6} x^{6}] [6 C_{0} + 6 C_{1} x + \dots \dots + 6 C_{6} x^{6}] \\ = (6 C_{0} \cdot 6 C_{6} + 6 C_{1} . 6 C_{5} + \dots \dots \dots \dots + 6 C_{6} \cdot 6 C_{0}) x^{6} + \dots \dots . \\ ∣ Coefficient of x^{6} in (1 + x)^{12} =^{12} C_{6} = 924 \end{array}$

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