How do you evaluate C(6, 12)?

To evaluate the combination \({}^{12}C_6\), we are dealing with a binomial coefficient, which is also referred to as "12 choose 6." This is the number of ways to choose 6 objects from a set of 12 distinct objects. The formula for the combination is:

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

In this case, \(n = 12\) and \(r = 6\). Substituting these values into the formula:

\[
{}^{12}C_6 = \frac{12!}{6!(12 - 6)!} = \frac{12!}{6!6!}
\]

Now, calculate the factorials:

- \(12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6! \)
- \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

Substitute the values into the equation:

\[
{}^{12}C_6 = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6!}{6! \times 6!} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6!}
\]

Now, simplify the expression by canceling out \(6!\) from the numerator and denominator:

\[
{}^{12}C_6 = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{720}
\]

Now, calculate the product of the numbers in the numerator:

\[
12 \times 11 = 132
\]
\[
132 \times 10 = 1320
\]
\[
1320 \times 9 = 11880
\]
\[
11880 \times 8 = 95040
\]
\[
95040 \times 7 = 665280
\]

Now, divide this by \(720\):

\[
\frac{665280}{720} = 924
\]

Therefore, the value of \({}^{12}C_6\) is **924**.