If you travel between Mumbai and Pune, you have to go through a 10 km long tunnel. Lately there has been a surge in the incidents of loot and plunder and many passengers have been harassed and injured inside the tunnel. Considering the enormity of the law and order problem, highway police decided to establish 4 security posts in order to ensure the safety of life and luggage of commuters. The minimum distance between any two posts must be n is 1 km. Each security post must be n km (n= o, 1, 2, 3,...) apart from entry/exit of the tunnel and none of them will be outside the tunnel. Find the number of ways of selecting the spots for upcoming security posts.

Options

A. 330
B. 110
C. 66
D. 120

Answer

To solve the problem of selecting spots for 4 security posts in a 10 km long tunnel, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Understand the Tunnel Length and Positions**:
The tunnel is 10 km long, which means it has positions from 0 km (entry) to 10 km (exit). This gives us a total of 11 positions (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).

2. **Determine the Number of Security Posts**:
We need to establish 4 security posts within the tunnel.

3. **Identify the Constraints**:
- The minimum distance between any two posts must be at least 1 km.
- Each post must be located within the tunnel (i.e., between 0 and 10 km).

4. **Calculate the Available Positions**:
Since we have 11 positions and we need to place 4 posts, we can use combinations to find the number of ways to choose 4 positions from these 11.

5. **Use the Combination Formula**:
The number of ways to choose k items from n items is given by the combination formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \( n = 11 \) (the total positions) and \( k = 4 \) (the number of posts).

6. **Calculate the Combinations**:
\[ \binom{11}{4} = \frac{11!}{4!(11-4)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} \]

7. **Simplify the Calculation**:
- Calculate the numerator:
\[ 11 \times 10 \times 9 \times 8 = 7920 \]
- Calculate the denominator:
\[ 4 \times 3 \times 2 \times 1 = 24 \]
- Now divide:
\[ \frac{7920}{24} = 330 \]

8. **Final Answer**:
Therefore, the number of ways to select the spots for the upcoming security posts is **330**.