Question

The students S1, S2, ......., S10 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 _________

Hint:

When distributing items into groups with "at least one" constraints, calculate total distributions and subtract empty-group cases using the Principle of Inclusion-Exclusion.

Answer 1

Correct Answer: 31650

Step 1: Let $|C| = r$. $r \in \{1, 2, 3\}$.
Step 2: For each $r$, choose students for $C$ in ${}^{10}C_r$ ways.
Step 3: Distribute remaining $10-r$ students into $A$ and $B$. Total $2^{10-r}$ ways.
Step 4: Subtract cases where $A$ or $B$ is empty: $2^{10-r} - 2$.
Step 5: Total $= \sum_{r=1}^3 {}^{10}C_r (2^{10-r} - 2) = 10(510) + 45(254) + 120(126) = 31650$.