Question

There are 3 students from section A, 5 from section B and 4 from section C. In how many ways can they occupy 6 seats such that the leftmost seat is occupied by a student of section A and the remaining seats are occupied by 3 students of section B and 2 students of section C?

• A. 10800
• B. 14400
• C. 18000
• D. 21600

Hint:

In seating arrangement problems, always separate the steps of selection and arrangement to avoid overcounting.

Answer 1

The Correct Option is D

Step 1: Choose the student for the leftmost seat from section A.
There are 3 students in section A, and the leftmost seat must be occupied by a student from section A.
Number of ways = \(3\)
Step 2: Select students from section B and section C.
We need to choose 3 students from section B out of 5:
\[ \binom{5}{3} = 10 \]
We need to choose 2 students from section C out of 4:
\[ \binom{4}{2} = 6 \]
Step 3: Arrange the selected students in the remaining 5 seats.
The selected 5 students (3 from B and 2 from C) can be arranged in the remaining 5 seats in:
\[ 5! = 120 \]
Step 4: Multiply all possible cases.
\[ 3 \times 10 \times 6 \times 120 = 21600 \]
Final Answer:
\[ \boxed{21600} \]