To evaluate the expression \({ }_4 P_2\), we use the formula for permutations given by \( nPr = \frac{n!}{(n-r)!} \).
Given \( n = 4 \) and \( r = 2 \), we can substitute these values into the formula:
[tex]\[ { }_4 P_2 = \frac{4!}{(4-2)!} \][/tex]
First, we need to calculate the factorials:
- \( 4! \) means the product of all positive integers up to 4, which is \( 4 \times 3 \times 2 \times 1 = 24 \).
- \((4-2)! = 2!\) means the product of all positive integers up to 2, which is \( 2 \times 1 = 2 \).
Now, we substitute these values back into the formula:
[tex]\[ { }_4 P_2 = \frac{24}{2} \][/tex]
Next, we perform the division:
[tex]\[ { }_4 P_2 = \frac{24}{2} = 12 \][/tex]
Therefore, the expression \({ }_4 P_2\) simplifies to:
[tex]\[ { }_4 P_2 = 12 \][/tex]
So, [tex]\({ }_4 P_2 = 12\)[/tex].
