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To find the value of the expression \( \binom{12}{5} \), we calculate the combination formula:

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

For \( \binom{12}{5} \), we substitute \( n = 12 \) and \( r = 5 \):

\[
\binom{12}{5} = \frac{12!}{5!(12 - 5)!} = \frac{12!}{5! \cdot 7!}
\]

To simplify, we calculate only up to \( 7! \) in \( 12! \):

\[
12 \times 11 \times 10 \times 9 \times 8 = 95,040
\]

Then we divide by \( 5! \):

\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

Now we calculate:

\[
\binom{12}{5} = \frac{95,040}{120} = 792
\]

The correct answer is:
**792**

Would you like a breakdown of each step, or have any questions?

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### Related Questions:
1. How is the formula for combinations derived?
2. What is the difference between permutations and combinations?
3. How would the calculation change if we were solving \( \binom{12}{4} \)?
4. How can factorials be simplified when computing combinations?
5. What are some real-life applications of combinations in probability?

**Tip:** In combinations, the order of selection doesn’t matter, while in permutations, it does.

What is the value of the expression 12C5?

Find the value of the combination 12C5 using the formula nCr = n! / (r!(n - r)!). This problem covers combinatorial concepts, factorial calculations, and is suitable for high school students learning about combinations.

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