Question

There are ten boys $B_{1}, B_{2}, \ldots, B_{10}$ and five girls $G_{1}$, $G _{2}, \ldots, G _{5}$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B_{1}$ and $B_{2}$ together should not be the members of a group, is________

Answer 1

$ n ( B )=10$
$ n ( a )=5$
The number of ways of forming a group of $3$ girls of $3$ boys.
$={ }^{10} C _{3} \times{ }^{5} C _{3} $
$=\frac{10 \times 9 \times 8}{3 \times 2} \times \frac{5 \times 4}{2}=1200$
The number of ways when two particular boys $B_{1}$ of $B_{2}$ be the member of group together
$={ }^{8} C _{1} \times{ }^{5} C _{3}=8 \times 10=80$
Number of ways when boys $B_{1}$ of $B_{2}$ hot in the same group together
$=1200 \times 80=1120$