Answer :
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.
[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.
