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

Options

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

Answer

To find the domain of the function \( f(x) = \binom{16-x}{2x-1} + P(20-3x, 4x-5) \), we need to ensure that the parameters for both the binomial coefficient and the permutation are valid.

### Step-by-Step Solution:

1. **Identify Conditions for Binomial Coefficient**:
The binomial coefficient \( \binom{n}{r} \) is defined when:
- \( n \geq 0 \)
- \( r \geq 0 \)
- \( n \geq r \)

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

We need to derive the inequalities:
- \( 16 - x \geq 0 \) implies \( x \leq 16 \)
- \( 2x - 1 \geq 0 \) implies \( x \geq \frac{1}{2} \)
- \( 16 - x \geq 2x - 1 \) implies:
\[ 16 - x \geq 2x - 1 \implies 16 + 1 \geq 3x \implies 17 \geq 3x \implies x \leq \frac{17}{3} \]

2. **Identify Conditions for Permutation**:
The permutation \( P(n, r) \) is defined when:
- \( n \geq 0 \)
- \( r \geq 0 \)
- \( n \geq r \)

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

We need to derive the inequalities:
- \( 20 - 3x \geq 0 \) implies \( x \leq \frac{20}{3} \)
- \( 4x - 5 \geq 0 \) implies \( x \geq \frac{5}{4} \)
- \( 20 - 3x \geq 4x - 5 \) implies:
\[ 20 + 5 \geq 7x \implies 25 \geq 7x \implies x \leq \frac{25}{7} \]

3. **Combine All Conditions**:
Now we have the following conditions:
- From the binomial coefficient:
- \( x \leq 16 \)
- \( x \geq \frac{1}{2} \)
- \( x \leq \frac{17}{3} \)
- From the permutation:
- \( x \leq \frac{20}{3} \)
- \( x \geq \frac{5}{4} \)
- \( x \leq \frac{25}{7} \)

4. **Determine the Overlapping Intervals**:
- The lower bounds are \( \frac{1}{2} \) and \( \frac{5}{4} \). The effective lower bound is \( \frac{5}{4} \) since it is greater.
- The upper bounds are \( 16, \frac{17}{3}, \frac{20}{3}, \frac{25}{7} \). The effective upper bound is \( \frac{20}{3} \) since it is the smallest.

5. **Final Domain**:
Thus, the domain of \( f(x) \) is:
\[ \frac{5}{4} \leq x \leq \frac{20}{3} \]
In interval notation, this is:
\[ \left[\frac{5}{4}, \frac{20}{3}\right] \]