What is the value of the expression ${ }_{12} C_5$ ?
The objective is to calculate the value of the combination \({}_{12}C_5\), which represents the number of ways to choose a subset of \(5\) items from a total set of \(12\) distinct items, where the order of selection does not matter. The general formula for combinations is:
\[{}_{n}C_k = \frac{n!}{k!(n-k)!}\]
Here, \(n = 12\) and \(k = 5\). Substituting these values into the formula gives:
\[{}_{12}C_5 = \frac{12!}{5!(12-5)!} = \frac{12!}{5!7!}\]
To simplify the expression \(\frac{12!}{5!7!}\), we expand the numerator \(12!\) until it reaches \(7!\) so that the \(7!\) terms in the numerator and denominator can be cancelled:
\[\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{5! \times 7!} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1}\]
Now, we calculate the denominator: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
The numerator is \(12 \times 11 \times 10 \times 9 \times 8 = 95,040\).
Finally, we perform the division:
\[\frac{95,040}{120} = 792\]
Comparing our calculated result of \(792\) with the provided options:
◯ \(33,264\)
◯ \(792\)
◯ \(95,040\)
◯ \(120\)
The correct option is \(792\). Thus, \({}_{12}C_5 = \boxed{792}\).
In real life, combinations are used whenever we need to select groups where the order doesn't matter. For instance, if a student needs to choose a committee of 5 members from a class of 12 students, the total number of different possible committees is given by \({}_{12}C_5 = 792\). This helps in organizing group projects, planning team rosters, or calculating the number of possible outcomes in various selection processes.