JEE Advance - Mathematics (2024 - Paper 1 Online - No. 11)
A group of 9 students, $s_1, s_2, \ldots, s_9$, is to be divided to form three teams $X, Y$, and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is ____________.
Answer
665
Explanation
$$\matrix{ x & y & z \cr 2 & 3 & 4 \cr {{{\overline S }_1}} & {{{\overline S }_2}} & {} \cr }$$
C-i) When x does not contain S$$_1$$, but contains S$$_2$$
$$\mathop {{}^7{C_1}}\limits_{for\,x} \times \mathop {{{7!} \over {3!4!}}}\limits_{for\,y,z} = 245$$
C-ii) When x does not contain $$\mathrm{S}_1, \mathrm{~S}_2$$ and y does not contain $$\mathrm{S}_2$$
i.e. $$\mathop {{}^7{C_2}}\limits_{for\,x} \times \mathop {{{6!} \over {3!3!}}}\limits_{for\,y,z} = 420$$
so total No. of ways 665
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