The domain of the function `f(x)=""^(16-x)C_(2x-1+^(20-3x)P_(4x-5)`, where the symbols have their usual meanings, is the set

Answer

To find the domain of the function \( f(x) = \binom{16-x}{2x-1} + P(20-3x, 4x-5) \), we need to analyze the conditions under which the binomial coefficient and the permutation are defined.

### Step 1: Analyze the Binomial Coefficient
The binomial coefficient \( \binom{n}{r} \) is defined when:
1. \( n \) is a natural number (non-negative integer).
2. \( r \) is a whole number (non-negative integer).
3. \( n \geq r \).

For our function, we have:
- \( n = 16 - x \)
- \( r = 2x - 1 \)

**Condition 1: \( 16 - x \geq 0 \)**
\[ 16 - x \geq 0 \implies x \leq 16 \]

**Condition 2: \( 2x - 1 \geq 0 \)**
\[ 2x - 1 \geq 0 \implies 2x \geq 1 \implies x \geq \frac{1}{2} \]

### Step 2: Analyze the Permutation
The permutation \( P(n, r) \) is defined when:
1. \( n \) is a natural number (non-negative integer).
2. \( r \) is a whole number (non-negative integer).
3. \( n \geq r \).

For our function, we have:
- \( n = 20 - 3x \)
- \( r = 4x - 5 \)

**Condition 3: \( 20 - 3x \geq 0 \)**
\[ 20 - 3x \geq 0 \implies 3x \leq 20 \implies x \leq \frac{20}{3} \approx 6.67 \]

**Condition 4: \( 4x - 5 \geq 0 \)**
\[ 4x - 5 \geq 0 \implies 4x \geq 5 \implies x \geq \frac{5}{4} = 1.25 \]

### Step 3: Combine Conditions
Now we combine all the conditions derived:
1. \( x \leq 16 \)
2. \( x \geq \frac{1}{2} \)
3. \( x \leq \frac{20}{3} \)
4. \( x \geq \frac{5}{4} \)

From conditions 2 and 4, we have:
\[ x \geq \frac{5}{4} \]

From conditions 1 and 3, we have:
\[ x \leq \frac{20}{3} \]

### Step 4: Determine the Domain
The combined inequalities give us:
\[ \frac{5}{4} \leq x \leq \frac{20}{3} \]

### Step 5: Identify Integer Values
Since \( x \) must be a natural number, we find the integer values within this range:
- The smallest integer \( \geq \frac{5}{4} \) is 2.
- The largest integer \( \leq \frac{20}{3} \) is 6.

Thus, the possible integer values for \( x \) are:
\[ x = 2, 3, 4, 5, 6 \]

### Final Domain
The domain of the function \( f(x) \) is:
\[ \{2, 3, 4, 5, 6\} \]