Question

The students $S_1,S_2,\dots ,S_{10}$ 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

Answer 1

Number of ways
$={}^{10}{C}_1[2^9-2]+{}^{10}{C}_2[2^8-2]+{}^{10}{C}_3[2^7-2]$
$=2^7[{}^{10}{C}_1\times 4+{}^{10}{C}_2\times 2+{}^{10}{C}_3]-20-90-240$
$=128[40+90+120]-350$
$=(128\times 250)-350$
$=10\left(3165\right)$
$=31650$