To identify \({}_{12}P_1\) using factorials, you need to understand that \({}_{n}P_{r}\) represents the number of permutations of \(r\) items from a set of \(n\) items. The formula for permutation is given by:
[tex]\[
{}_{n}P_{r} = \frac{n!}{(n-r)!}
\][/tex]
In this case, \(n = 12\) and \(r = 1\). Substituting these values into the permutation formula, we get:
[tex]\[
{}_{12}P_{1} = \frac{12!}{(12-1)!}
\][/tex]
This simplifies to:
[tex]\[
{}_{12}P_{1} = \frac{12!}{11!}
\][/tex]
Now, knowing that \(n!\) (read as "n factorial") is the product of all positive integers up to \(n\), we can evaluate \(12!\) and \(11!\):
- \(12!\) (12 factorial) is the product of all integers from 1 to 12.
- \(11!\) (11 factorial) is the product of all integers from 1 to 11.
From the calculations, we have:
[tex]\[
12! = 479001600
\][/tex]
[tex]\[
11! = 39916800
\][/tex]
Next, we need to compute the quotient of these factorials:
[tex]\[
\frac{12!}{11!} = \frac{479001600}{39916800} = 12.0
\][/tex]
Therefore, \({}_{12}P_{1}\) is:
[tex]\[
{}_{12}P_{1} = 12.0
\][/tex]
This means there are 12 different ways to arrange 1 item out of a set of 12 items.
