First slide

Combinations

Question

A committee consisting of at least three members is to be formed from a group of $6$ boys and $6$ girls such that it always has a boy and a girl. The number of ways to form such committee is:

Moderate

A

$2^{12} - 2^{7} - 13$
B

$2^{11} - 2^{6} - 13$
C

$2^{12} - 2^{7} - 35$
D

$2^{11} - 2^{7} - 35$

Solution

. Required number of ways
$\begin{array}{l} = (^{6} C_{1} +^{6} C_{2} + \dots +^{6} C_{6}) (^{6} C_{1} +^{6} C_{2} + \dots +^{6} C_{6}) - \\ (^{6} C_{1}) (^{6} C_{1}) \\ = (2^{6} - 1) (2^{6} - 1) - 36 = 2^{12} - 2^{7} - 35 \end{array}$

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