Evaluate the expression. Round the answer to the nearest whole number.
12 Upper C 812C8
Question content area bottom
Part 1
A.9 comma 979 comma 200
9 comma 979 comma 2009,979,200
B.48
4848
C.11 comma 880
11 comma 88011,880
D.495
495495

The expression "12 choose 8" (denoted as \( \binom{12}{8} \)) is a combination calculation. It calculates how many ways you can choose 8 items from a set of 12 without regard to the order.

The formula for combinations is:

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

For this problem, \( n = 12 \) and \( r = 8 \). Plugging in the values:

\[
\binom{12}{8} = \frac{12!}{8!(12 - 8)!} = \frac{12!}{8!4!}
\]

Now let's compute this.

\[
12! = 12 \times 11 \times 10 \times 9 \times 8!
\]

We can cancel out the \( 8! \) terms in the numerator and denominator:

\[
\frac{12!}{8!4!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1}
\]

Simplifying this:

\[
= \frac{12 \times 11 \times 10 \times 9}{24} = \frac{11880}{24} = 495
\]

Thus, \( \binom{12}{8} = 495 \).

The correct answer is **D. 495**.

Would you like more details or have any questions about this process?

Here are 5 related questions to expand your understanding:
1. How is the combination formula different from the permutation formula?
2. What does the factorial operation represent in mathematics?
3. How would you calculate \( \binom{12}{4} \) and why is it the same as \( \binom{12}{8} \)?
4. Can you explain a real-world scenario where combinations are used?
5. How does the order of selection affect the difference between combinations and permutations?

**Tip:** When calculating combinations, you can simplify by canceling out common factorial terms to make the calculations easier.

Evaluate the expression. Round the answer to the nearest whole number.
12 Upper C 812C8
A.9 comma 979 comma 200
B.48
C.11 comma 880
D.495

Learn how to calculate combinations using the formula \( \binom{12}{8} \), which evaluates to 495. This problem covers the combination formula, factorial simplification, and related combinatorial concepts. Suitable for students in Grades 9-12.

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