1. **Stating the problem:** Find the value of $\binom{n}{0}$ for each positive integer $n$. 2. **Formula and explanation:** The binomial coefficient $\binom{n}{k}$ is defined as: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ is the factorial of $n$. 3. **Applying the formula for $k=0$:** $$\binom{n}{0} = \frac{n!}{0! \cdot (n-0)!} = \frac{n!}{1 \cdot n!} = 1$$ 4. **Interpretation:** Choosing zero elements from $n$ elements can only be done in exactly one way — by choosing nothing. 5. **Final answer:** $$\binom{n}{0} = 1$$ Therefore, the correct choice is C) 1.