Question

The number of elements in the set \( \left\{ x \in \mathbb{N} : \binom{20 - 2x}{x - 3} \in \mathbb{N} \right\} \) is

• A. \( 3 \)
• B. \( 4 \)
• C. \( \mathbf{5} \)
• D. \( 6 \)

Hint:

For binomial coefficients to be natural numbers, the top must be greater than or equal to the bottom and both must be non-negative integers.

Answer 1

The Correct Option is C

We are given \( \binom{20 - 2x}{x - 3} \in \mathbb{N} \). This binomial coefficient is defined only when: \[ 0 \leq x - 3 \leq 20 - 2x \Rightarrow x \geq 3 \text{ and } x \leq \frac{23}{3} \Rightarrow x \in \{3, 4, 5, 6, 7\} \] So, the number of such values is \( 5 \).