Three schools send 2, 4, and 6 students,respe | vedclass

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(C) The number of students from the three schools are $2, 4,$ and $6$. Since each room must contain $2$ students from the same school,we divide the st

Vedclass · Accepted Answer

$32400$

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(C) The number of students from the three schools are $2, 4,$ and $6$. Since each room must contain $2$ students from the same school,we divide the students into groups of $2$: School $1$ has $1$ group of $2$ students. School $2$ has $2$ groups of $2$ students. School $3$ has $3$ groups of $2$ students. Total number of groups $= 1 + 2 + 3 = 6$ groups. First,we arrange these $6$ groups into the $6$ distinct rooms,which can be done in $6!$ ways. Next,we consider the internal arrangements of the students within the schools: For school $2$,the $4$ students are divided into $2$ groups of $2$. The number of ways to do this is $\frac{4!}{2! 	imes 2! 	imes 2!} = 3$. For school $3$,the $6$ students are divided into $3$ groups of $2$. The number of ways to do this is $\frac{6!}{2! 	imes 2! 	imes 2! 	imes 3!} = 15$. Total ways $= 6! 	imes 3 	imes 15 = 720 	imes 45 = 32400$.

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