Question:
The number of elements in the set \( \left\{ x \in \mathbb{N} : \binom{20 - 2x}{x - 3} \in \mathbb{N} \right\} \) is
The number of elements in the set \( \left\{ x \in \mathbb{N} : \binom{20 - 2x}{x - 3} \in \mathbb{N} \right\} \) is
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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.
Updated On: May 15, 2025
- \( 3 \)
- \( 4 \)
- \( \mathbf{5} \)
- \( 6 \)
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The Correct Option is C
Solution and Explanation
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 \( 5 \).
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