There are 5 historical moments, 6 gardens and 7 shopping malls in the city. In how many ways a tourist can visit the city, if he visits at least one shopping mall?

Options

A. `2^5 .2^6 . ( 2^7)`
B. `2^4 . 2^6 (2^7 -1)`
C. `2^5 . 2^6 (2^6 -1)`
D. None of these

Answer

To solve the problem of how many ways a tourist can visit the city while ensuring that he visits at least one shopping mall, we can break down the solution step by step.

### Step-by-Step Solution:

1. **Identify the Choices for Historical Moments:**
- There are 5 historical moments. For each historical moment, the tourist has 2 choices: either to visit it or not to visit it.
- Therefore, the total choices for historical moments can be calculated as:
\[ 2^5 \]

2. **Identify the Choices for Gardens:**
- Similarly, there are 6 gardens. For each garden, the tourist also has 2 choices: to visit or not to visit.
- Thus, the total choices for gardens can be calculated as:
\[ 2^6 \]

3. **Identify the Choices for Shopping Malls:**
- There are 7 shopping malls. For each shopping mall, the tourist has 2 choices: to visit or not to visit.
- However, since the tourist must visit at least one shopping mall, we need to calculate the total choices while excluding the scenario where he visits none.
- The total choices for shopping malls without any restrictions is:
\[ 2^7 \]
- To find the number of ways to visit at least one shopping mall, we subtract the scenario where he visits none (which is 1 way):
\[ 2^7 - 1 \]

4. **Combine All Choices:**
- Now, we can combine all the choices for historical moments, gardens, and shopping malls:
\[ \text{Total Ways} = (2^5) \times (2^6) \times (2^7 - 1) \]

5. **Calculate the Total Ways:**
- First, calculate \(2^5\), \(2^6\), and \(2^7\):
\[ 2^5 = 32, \quad 2^6 = 64, \quad 2^7 = 128 \]
- Therefore, \(2^7 - 1 = 128 - 1 = 127\).
- Now substitute these values back into the total ways equation:
\[ \text{Total Ways} = 32 \times 64 \times 127 \]

6. **Final Calculation:**
- Calculate \(32 \times 64\):
\[ 32 \times 64 = 2048 \]
- Now multiply by 127:
\[ 2048 \times 127 = 260096 \]

### Conclusion:
The total number of ways a tourist can visit the city while ensuring that he visits at least one shopping mall is **260096**.