The students S_1,,S_10 are to be divided into 3 groups A, B and C such that each group has…

Question

The students $S_1,\ldots,S_{10}$ are to be divided into 3 groups A, B and C such that each group has at least one student and group C has at most 3 students. Then the total number of possibilities of forming such groups is .........

Answer: 31650

Accepted Answer

Case C has 1 student: $\binom{10}{1}$ ways to pick, remaining 9 split into non-empty A,B: $2^9-2$ ways.

Case C has 2 students: $\binom{10}{2}\times(2^8-2)$

Case C has 3 students: $\binom{10}{3}\times(2^7-2)$

$$\text{Total} = \binom{10}{1}(2^9-2) + \binom{10}{2}(2^8-2) + \binom{10}{3}(2^7-2)$$
$$= 10\times510 + 45\times254 + 120\times126 = 5100+11430+15120 = 31650$$

Solutions · 1