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6.Permutation and Combination
hard
The students $S _{1}, S _{2}, \ldots \ldots, S _{10}$ are to be divided into $3$ groups $A , B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is ........ .
A
$63300$
B
$42580$
C
$15325$
D
$31650$
(JEE MAIN-2021)
Solution
If group $C$ has one student then number of groups
${ }^{10} C _{1}\left[2^{9}-2\right]=5100$
If group $C$ has two students then number of groups
${ }^{10} C _{2}\left[2^{8}-2\right]=11430$
If group $C$ has three students then number of groups
$={ }^{10} C _{3} \times\left[2^{7}-2\right]=15120$
So total groups $=31650$
Standard 11
Mathematics
