A group of 9 students, $\mathrm{s}_1, \mathrm{~s}_2, \ldots \ldots, \mathrm{s}_9$, is to be divided to from…
A group of 9 students, $\mathrm{s}_1, \mathrm{~s}_2, \ldots \ldots, \mathrm{s}_9$, is to be divided to from three teams $\mathrm{X}, \mathrm{Y}$, and $\mathrm{Z}$ of sizes 2,3 , and 4 , respectively. Suppose that $s_1$ cannot be selected for the team $\mathrm{X}$, and $\mathrm{s}_2$ cannot be selected for the team Y. Then the number of ways to from such teams, is
Solution
$\begin{aligned} x & y & z \\ 2 & 3 & 4 \end{aligned}$
C-i) when $x$ does not contain $S_1$, but contains $S_2$
$\underset{\text{(for }x)}{7 C_1} \times \underset{\text{(for }y,z)}{\frac{7!}{3!4!}}=245$
C-ii) When $x$ does not contain $S_1, S_2$ and $y$ does not contain $S_2$ i.e. $\underset{\text{(for }x)}{7 C_2} \times \underset{\text{(for }y,z)}{\frac{6!}{3!3!}}=420$
so total No. of ways $665$