There are 10 seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is:

Options

A. `""^(7)C_(4)`
B. `4.""^(7)P_(3)`
C. `""^(7)C_(3)*4!`
D. `840`

Answer

To solve the problem of arranging 4 persons in 10 seats such that no two persons sit side by side, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Seats and Persons**:
- We have a total of 10 seats and we need to occupy 4 of them with persons.

2. **Calculate the Remaining Seats**:
- After occupying 4 seats, the remaining seats are:
\[ 10 - 4 = 6 \text{ seats} \]

3. **Understanding the Arrangement**:
- To ensure that no two persons sit next to each other, we need to create gaps between the occupied seats.
- If we occupy 4 seats, we will create gaps around them. For 4 persons, we will have:
- 1 gap before the first person
- 1 gap between each pair of persons (3 gaps)
- 1 gap after the last person
- This gives us a total of:
\[ 1 + 3 + 1 = 5 \text{ gaps} \]
- Hence, we need to arrange 4 persons in these 5 gaps.

4. **Calculate the Available Positions**:
- The total number of positions available for the 4 persons is equal to the remaining seats plus the gaps created, which is:
\[ 6 \text{ (remaining seats)} + 1 \text{ (gap before the first person)} + 1 \text{ (gap after the last person)} = 8 \text{ positions} \]

5. **Choosing Positions for Persons**:
- We can choose 4 positions from these 8 available positions. The number of ways to choose 4 positions from 8 is given by the combination formula \( \binom{n}{r} \):
\[ \binom{8}{4} \]

6. **Arranging the Persons**:
- Once we have chosen the 4 positions, the 4 persons can be arranged in these positions. The number of ways to arrange 4 persons is given by \( 4! \) (factorial of 4):
\[ 4! = 24 \]

7. **Total Arrangements**:
- Therefore, the total number of ways to arrange the 4 persons such that no two sit next to each other is:
\[ \text{Total Ways} = \binom{8}{4} \times 4! \]

8. **Calculating the Values**:
- First, calculate \( \binom{8}{4} \):
\[ \binom{8}{4} = \frac{8!}{4! \cdot (8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]
- Now, calculate the total arrangements:
\[ \text{Total Ways} = 70 \times 24 = 1680 \]

### Final Answer:
The number of ways of arranging 4 persons so that no two persons sit side by side is **1680**.