The range of function `f(x)=^(7-x)P_(x-3)i s` (a) {1,2,3} (b) `{1,2,3,4,5,6}` `(c){1,2,3,4}` (d) `{1,2,3,4,5}`

Options

A. `{1,2,3}`
B. `{1,2,3,4,5,6}`
C. `{1,2,3,4}`
D. `{1,2,3,4,5}`

Explanation

To find the range of the function \( f(x) = (7 - x) P (x - 3) \), we will follow these steps:

### Step 1: Understand the Permutation Formula
The permutation \( nP_r \) is defined as:
\[ nP_r = \frac{n!}{(n - r)!} \]
In our case, we have:
\[ f(x) = (7 - x) P (x - 3) = \frac{(7 - x)!}{(7 - x - (x - 3))!} = \frac{(7 - x)!}{(10 - 2x)!} \]

### Step 2: Determine the Validity Conditions
For the permutation to be valid:
1. \( n = 7 - x \) must be non-negative, which gives:
\[ 7 - x \geq 0 \implies x \leq 7 \]
2. \( r = x - 3 \) must also be non-negative, which gives:
\[ x - 3 \geq 0 \implies x \geq 3 \]
3. Additionally, \( n \) must be greater than or equal to \( r \):
\[ 7 - x \geq x - 3 \implies 10 \geq 2x \implies x \leq 5 \]

### Step 3: Combine the Conditions
From the conditions derived:
- \( x \leq 7 \)
- \( x \geq 3 \)
- \( x \leq 5 \)

The combined valid range for \( x \) is:
\[ 3 \leq x \leq 5 \]

### Step 4: Evaluate the Function at Integer Values
The integer values of \( x \) in the range are \( 3, 4, 5 \).

1. For \( x = 3 \):
\[ f(3) = (7 - 3) P (3 - 3) = 4 P 0 = 1 \]
2. For \( x = 4 \):
\[ f(4) = (7 - 4) P (4 - 3) = 3 P 1 = 3 \]
3. For \( x = 5 \):
\[ f(5) = (7 - 5) P (5 - 3) = 2 P 2 = 2 \]

### Step 5: Determine the Range
The values obtained for \( f(x) \) are:
- \( f(3) = 1 \)
- \( f(4) = 3 \)
- \( f(5) = 2 \)

Thus, the range of \( f(x) \) is:
\[ \{1, 2, 3\} \]

### Conclusion
The correct option for the range of the function \( f(x) \) is:
**(a) {1, 2, 3}**

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