Question

If ⁿC_r denotes the number of combinations of n distinct things taken r at a time, then the domain of the function g (x)= ^(16-x)C_(2x-1)  is

• A. {1,2,3,4,5}
• B. {0,1,2,3.4}
• C. ⦰
• D. {0}

Answer 1

The Correct Option is A

To solve the problem, we need to determine the domain of the function \( g(x) = \binom{16 - x}{2x - 1} \), where the binomial coefficient is defined only when the upper value is a non-negative integer and the lower value is a non-negative integer ≤ upper value.

1. Understand the Condition for Combinations:
The function involves a binomial coefficient \( \binom{n}{r} \), which is only defined when:
- \( n \in \mathbb{Z}_{\geq 0} \) (i.e., n is a non-negative integer)
- \( r \in \mathbb{Z}_{\geq 0} \) and \( r \leq n \)

2. Apply to the Given Function:
We are given \( g(x) = \binom{16 - x}{2x - 1} \)

3. Set Conditions for Domain:
For the binomial to be defined:
- \( 16 - x \in \mathbb{Z}_{\geq 0} \Rightarrow x \leq 16 \)
- \( 2x - 1 \in \mathbb{Z}_{\geq 0} \Rightarrow x \geq \frac{1}{2} \)
- Also, \( 2x - 1 \leq 16 - x \Rightarrow 3x \leq 17 \Rightarrow x \leq \frac{17}{3} \approx 5.67 \)

4. Find Integer Values Satisfying All Conditions:
We need integer values of \( x \) such that:
- \( \frac{1}{2} \leq x \leq \frac{17}{3} \)
So, possible integer values are: \( x = 1, 2, 3, 4, 5 \)

Final Answer:
The domain of the function is \( \{1, 2, 3, 4, 5\} \)