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Q.
There are 6 boys and 6 girls. The number of ways to choose atleast 2 persons so that the combination include at least a boy and a girl is
Permutations and Combinations
Solution:
Number of ways = Total ways - no boy - no girl $+1 $
$=\left({ }^6 C _0+{ }^6 C _1+\ldots \ldots+{ }^6 C _6\right) \times\left({ }^6 C _0+{ }^6 C _1+\ldots \ldots{ }^6 C _6\right)$
$ \quad{ }^6 C _0 \times\left({ }^6 C _0+{ }^6 C _1+{ }^6 C _2+\ldots \ldots \cdot{ }^6 C _6\right)-{ }^6 C _0 \times\left({ }^6 C _0+\ldots \ldots+{ }^6 C _0\right)+1 $
$=2^{12}-2^6-2^6+1=2^{12}-2^7+1$
Aliter:
Required number of ways $=\left(2^6-1\right)\left(2^6-1\right)=2^{12}-2^7+1$