Question

In how many number of ways can 10 students be divided into three teams, one containing four students and the other three?

• A. 400
• B. 700
• C. 1050
• D. 2100

Hint:

When two groups have same size, divide by factorial of identical groups to avoid overcounting.

Answer 1

The Correct Option is D

Step 1: Understand grouping.
We divide 10 students into:
- Team 1: 4 students
- Team 2: 3 students
- Team 3: 3 students
Step 2: Choose 4 students for the first team.
\[ \binom{10}{4} \]
Step 3: From remaining 6 students choose 3 for second team.
\[ \binom{6}{3} \]
Step 4: Remaining 3 automatically form third team.
So total arrangements:
\[ \binom{10}{4}\binom{6}{3} \]
Step 5: Divide by \(2!\) because two teams of 3 are identical.
\[ \text{Ways}=\frac{\binom{10}{4}\binom{6}{3}}{2!} \]
Step 6: Compute values.
\[ \binom{10}{4}=210,\quad \binom{6}{3}=20 \]
\[ \text{Ways}=\frac{210\times 20}{2}=2100 \]
Final Answer:
\[ \boxed{2100} \]