John parked his car among 10 cars in a row in a mall parking, not at either end, which has a capacity of 11 cars. In how many ways is it possible that on his return from the mall, he finds that exactly 5 (including his car) of the 11 places are still occupied, if both neighbouring parking plots around his car are empty?

To solve this problem, let's first understand the constraints and the situation:

1. **Total parking spots**: 11.
2. **John's car**: parked among the 10 cars in a row, but not at either end. This means John's car is in one of the positions from 2 to 10.
3. **Exactly 5 cars, including John's, are still parked**: After returning from the mall, 5 spots (including John's) are occupied.
4. **Both neighboring spots around John's car are empty**: This means the two spots adjacent to John's car must be unoccupied.

### Step-by-Step Solution

#### Step 1: Determine John's Position

John's car is parked in a position from 2 to 10 (not at either end). Let’s denote John’s car position by \( j \) where \( j = 2, 3, \ldots, 10 \).

#### Step 2: Neighboring Spots Must Be Empty

Given that both neighboring spots around John's car are empty, for any position \( j \), the spots \( j-1 \) and \( j+1 \) are empty.

This means we need to place John's car at a position \( j \) such that both spots \( j-1 \) and \( j+1 \) are vacant.

#### Step 3: Choosing the Remaining Occupied Spots

We need exactly 5 cars in total to be parked, including John's. So, there are 4 more cars to be placed in the remaining 8 spots (11 total spots minus John's car and the two empty spots next to it).

The number of ways to choose these 4 spots from the remaining 8 is given by:

\[
\binom{8}{4}
\]

#### Step 4: Total Number of Valid Configurations

John's car can be in any of the positions \( j = 2, 3, \ldots, 10 \), which gives us 9 possible choices for \( j \).

For each choice of \( j \), the number of ways to choose 4 more cars from the remaining 8 spots is:

\[
\binom{8}{4} = 70
\]

Thus, the total number of ways in which exactly 5 of the 11 places are occupied, and both neighbors of John's car are empty, is:

\[
9 \times 70 = 630
\]

### Final Answer

The total number of ways is **630**.

Would you like more details on any part of the solution? Here are some related questions you might consider:

1. What if John parked his car at one of the ends?
2. How would the solution change if there were more or fewer cars parked initially?
3. What if exactly 6 spots were occupied instead of 5?
4. How would the number of combinations change if the parking spots were circular (like a loop)?
5. How do combinations and permutations differ in problems like these?

**Tip:** Problems involving constraints often benefit from breaking them down into smaller parts and considering each constraint separately before combining them!

Explore how to determine the number of ways John can find exactly 5 occupied parking spots when returning to the mall. Includes detailed steps on using combinatorial methods and binomial coefficients. Suitable for high school students.

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