If `15! =2^alpha* 3^beta*5^gamma*7^delta*11^theta * 13^phi`, then the value of `alpha-beta+gamma-delta+theta-phi` is:

Options

A. (a) `4`
B. (b) `6`
C. (c) `8`
D. (d) `10`

Answer

To solve the problem, we need to find the values of \( \alpha, \beta, \gamma, \delta, \theta, \phi \) in the prime factorization of \( 15! \) and then compute \( \alpha - \beta + \gamma - \delta + \theta - \phi \).

### Step-by-Step Solution:

1. **Calculate \( 15! \)**:
\[ 15! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11 \times 12 \times 13 \times 14 \times 15 \]

2. **Identify the prime factorization of each number**:
- \( 1 \) contributes nothing.
- \( 2 = 2^1 \)
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- \( 10 = 2^1 \times 5^1 \)
- \( 11 = 11^1 \)
- \( 12 = 2^2 \times 3^1 \)
- \( 13 = 13^1 \)
- \( 14 = 2^1 \times 7^1 \)
- \( 15 = 3^1 \times 5^1 \)

3. **Count the total powers of each prime**:
- **For \( 2 \)**:
- From \( 2: 1 \)
- From \( 4: 2 \)
- From \( 6: 1 \)
- From \( 8: 3 \)
- From \( 10: 1 \)
- From \( 12: 2 \)
- From \( 14: 1 \)
- Total: \( 1 + 2 + 1 + 3 + 1 + 2 + 1 = 11 \)
- So, \( \alpha = 11 \)

- **For \( 3 \)**:
- From \( 3: 1 \)
- From \( 6: 1 \)
- From \( 9: 2 \)
- From \( 12: 1 \)
- From \( 15: 1 \)
- Total: \( 1 + 1 + 2 + 1 + 1 = 6 \)
- So, \( \beta = 6 \)

- **For \( 5 \)**:
- From \( 5: 1 \)
- From \( 10: 1 \)
- From \( 15: 1 \)
- Total: \( 1 + 1 + 1 = 3 \)
- So, \( \gamma = 3 \)

- **For \( 7 \)**:
- From \( 7: 1 \)
- From \( 14: 1 \)
- Total: \( 1 + 1 = 2 \)
- So, \( \delta = 2 \)

- **For \( 11 \)**:
- From \( 11: 1 \)
- Total: \( 1 \)
- So, \( \theta = 1 \)

- **For \( 13 \)**:
- From \( 13: 1 \)
- Total: \( 1 \)
- So, \( \phi = 1 \)

4. **Calculate \( \alpha - \beta + \gamma - \delta + \theta - \phi \)**:
\[ \alpha - \beta + \gamma - \delta + \theta - \phi = 11 - 6 + 3 - 2 + 1 - 1 \]
\[ = 11 - 6 = 5 \]
\[ = 5 + 3 = 8 \]
\[ = 8 - 2 = 6 \]
\[ = 6 + 1 = 7 \]
\[ = 7 - 1 = 6 \]

5. **Final Answer**:
The value of \( \alpha - \beta + \gamma - \delta + \theta - \phi \) is \( 6 \).