Question:
The students $\mathrm{S}_{1}, \mathrm{~S}_{2}, \ldots . ., \mathrm{S}_{10}$ are to be divided into 3 groups $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ 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
Solution:
If group $\mathrm{C}$ has one student then number of groups
${ }^{10} \mathrm{C}_{1}\left[2^{9}-2\right]=5100$
If group $C$ has two students then number of groups
${ }^{10} \mathrm{C}_{2}\left[2^{8}-2\right]=11430$
If group $C$ has three students then number of groups
$={ }^{10} \mathrm{C}_{3} \times\left[2^{7}-2\right]=15120$
So total groups $=31650$