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?

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.