The domain of definition of the function `f(x)=(7- x)P_(x-3)` , is

Options

A. [3,7]
B. {3,4,5,6,7}
C. {3,4,5}
D. none of these

Answer

To find the domain of the function \( f(x) = (7 - x) P_{(x - 3)} \), we need to analyze the conditions under which the permutation \( P_{(n, r)} \) is defined.

### Step-by-Step Solution:

1. **Understanding Permutation**:
The permutation \( P_{n, r} \) is defined as the number of ways to arrange \( r \) objects out of \( n \) objects. The conditions for this to be defined are:
- \( n \) must be a natural number (positive integers: 1, 2, 3, ...).
- \( r \) must be a whole number (non-negative integers: 0, 1, 2, ...).
- \( n \) must be greater than or equal to \( r \) (i.e., \( n \geq r \)).

2. **Identifying \( n \) and \( r \)**:
In our function, we have:
- \( n = 7 - x \)
- \( r = x - 3 \)

3. **Condition for \( n \)**:
Since \( n \) must be a natural number:
\[ 7 - x > 0 \implies x < 7 \]
Therefore, \( x \) must be less than 7.

4. **Condition for \( r \)**:
Since \( r \) must be a whole number:
\[ x - 3 \geq 0 \implies x \geq 3 \]
Therefore, \( x \) must be greater than or equal to 3.

5. **Condition for \( n \geq r \)**:
We also need to satisfy the condition \( n \geq r \):
\[ 7 - x \geq x - 3 \]
Rearranging this gives:
\[ 7 + 3 \geq 2x \implies 10 \geq 2x \implies 5 \geq x \]
Therefore, \( x \) must be less than or equal to 5.

6. **Combining Conditions**:
Now we have three conditions:
- \( x < 7 \)
- \( x \geq 3 \)
- \( x \leq 5 \)

The combined conditions give us:
\[ 3 \leq x \leq 5 \]

7. **Conclusion**:
The domain of the function \( f(x) \) is the interval:
\[ [3, 5] \]

### Final Answer:
The domain of definition of the function \( f(x) = (7 - x) P_{(x - 3)} \) is \( [3, 5] \).