Number of ways in which 4 students can sit in 7 chair in a row, if there is no empty chair between any two students is :

Options

A. 24
B. 28
C. 72
D. 96

Answer

To solve the problem of how many ways 4 students can sit in 7 chairs in a row without any empty chair between them, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
We have 4 students (let's call them S1, S2, S3, S4) and 7 chairs. The condition states that there should be no empty chair between any two students. This means that the 4 students must sit together as a block.

2. **Treating Students as a Block**:
Since the 4 students must sit together, we can treat them as a single unit or block. This block will occupy 4 consecutive chairs.

3. **Finding Possible Positions for the Block**:
We need to determine how many ways we can position this block of 4 students within the 7 chairs. The block can start at the following positions:
- Position 1 (occupying chairs 1, 2, 3, 4)
- Position 2 (occupying chairs 2, 3, 4, 5)
- Position 3 (occupying chairs 3, 4, 5, 6)
- Position 4 (occupying chairs 4, 5, 6, 7)

Thus, there are 4 possible positions for the block of 4 students.

4. **Arranging the Students within the Block**:
Within this block, the 4 students can be arranged among themselves in different ways. The number of arrangements of 4 students is given by the factorial of the number of students, which is \(4!\).

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

5. **Calculating the Total Arrangements**:
To find the total number of arrangements, we multiply the number of ways to position the block by the number of arrangements of the students within that block.

\[ \text{Total arrangements} = \text{Number of positions} \times \text{Arrangements within the block} = 4 \times 24 = 96 \]

### Final Answer:
The total number of ways in which 4 students can sit in 7 chairs in a row, without any empty chair between them, is **96**.

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