Yes, your calculation is correct! The binomial coefficient \(\binom{7}{2}\) is computed as follows:

\[
\binom{7}{2} = \frac{7 \times 6}{2} = \frac{42}{2} = 21
\]

The binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements, and it is calculated using the formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

For \(\binom{7}{2}\), the simplified form works out to 21 as you've shown.

Would you like more details or have any further questions?

Here are 5 related questions to expand on this topic:

1. What is the value of \(\binom{8}{3}\)?
2. How does \(\binom{n}{k}\) relate to Pascal's Triangle?
3. Can you explain the symmetry property \(\binom{n}{k} = \binom{n}{n-k}\)?
4. How does the binomial theorem use binomial coefficients?
5. What is the connection between binomial coefficients and combinations in probability?

**Tip:** In combinatorics, \(\binom{n}{k}\) is often referred to as "n choose k," which represents choosing \(k\) items from a group of \(n\) without regard to order.

Learn how to calculate the binomial coefficient \binom{7}{2} = 21 using the combination formula. This solution covers combinatorics concepts, such as combinations and binomial coefficients, and provides a step-by-step breakdown. Ideal for students in Grades 9-12.

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