[ \binom{7}{2} \times 2! = \frac{7!}{2!(7-2)!} \times 2 = \frac{7 \times 6}{2 \times 1} \times 2 = 21 \times 2 = 42 ]

The calculation you've provided is an evaluation of a binomial coefficient \(\binom{7}{2}\), followed by multiplication by \(2!\). Let's break down the steps:

1. **The binomial coefficient** \(\binom{7}{2}\) is calculated as:

\[
\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21
\]

2. **Multiplying by \(2!\):**

Since \(2! = 2\), the next step is multiplying the binomial result by \(2\):

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

So, the final result is **42**.

Would you like further details or any questions on this process?

Here are 5 related questions to deepen understanding:

1. How do you calculate the binomial coefficient \(\binom{n}{r}\) in general?
2. What does the binomial coefficient represent in combinatorics?
3. What are factorials, and how do they relate to binomial coefficients?
4. How can we apply the binomial theorem to expand algebraic expressions?
5. How does changing the value of \(n\) or \(r\) in \(\binom{n}{r}\) affect the result?

**Tip**: Binomial coefficients are widely used in probability and combinatorics, representing the number of ways to choose \(r\) objects from \(n\) distinct ones.

\binom{7}{2} \times 2! = \frac{7!}{2!(7-2)!} \times 2 = \frac{7 \times 6}{2 \times 1} \times 2 = 21 \times 2 = 42

Explore how to calculate the binomial coefficient \binom{7}{2} and multiply by factorial 2! with step-by-step solutions. Understand key concepts in combinatorics and factorials, suitable for students in Grades 9-12.

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