A zoo has 25 zebras, 14 girrafes, 16 lions and 2 tigers. In how many ways can a tourist visit these animals so that he must see at least one tiger.

Answer

To solve the problem of how many ways a tourist can visit the animals in the zoo while ensuring that he sees at least one tiger, we can follow these steps:

### Step 1: Identify the total number of animals
The zoo has:
- 25 zebras
- 14 giraffes
- 16 lions
- 2 tigers

### Step 2: Calculate the total ways to see each type of animal
For each type of animal, the tourist can either choose to see the animal or not see it. Therefore, for each animal type, the number of ways to see them can be calculated as follows:

- For zebras: \(2^{25}\) (each zebra can be seen or not seen)
- For giraffes: \(2^{14}\)
- For lions: \(2^{16}\)
- For tigers: \(2^{2}\)

### Step 3: Calculate the total ways to see all animals without any restriction
Now, we can multiply the ways to see each type of animal:

\[ \text{Total ways} = 2^{25} \times 2^{14} \times 2^{16} \times 2^{2} \]

### Step 4: Simplify the expression
Using the property of exponents, we can combine the powers:

\[ \text{Total ways} = 2^{(25 + 14 + 16 + 2)} = 2^{57} \]

### Step 5: Calculate the ways to see animals without seeing any tigers
If the tourist does not see any tigers, then the number of ways to see the other animals would be:

- For zebras: \(2^{25}\)
- For giraffes: \(2^{14}\)
- For lions: \(2^{16}\)
- For tigers: \(2^{0}\) (since he is not seeing any tigers)

Thus, the total ways without seeing any tigers is:

\[ \text{Ways without tigers} = 2^{25} \times 2^{14} \times 2^{16} \times 2^{0} = 2^{(25 + 14 + 16 + 0)} = 2^{55} \]

### Step 6: Calculate the ways to see at least one tiger
To find the number of ways to see at least one tiger, we subtract the ways without seeing any tigers from the total ways:

\[ \text{Ways with at least one tiger} = \text{Total ways} - \text{Ways without tigers} \]

\[ = 2^{57} - 2^{55} \]

### Step 7: Factor out the common term
We can factor out \(2^{55}\):

\[ = 2^{55}(2^{2} - 1) = 2^{55}(4 - 1) = 2^{55} \times 3 \]

### Final Answer
Thus, the total number of ways the tourist can visit the animals in the zoo while ensuring that he sees at least one tiger is:

\[ \text{Final Answer} = 3 \times 2^{55} \]