To calculate [tex]\( {}_6P_6 \)[/tex], we use the formula for permutations:
[tex]\[
{}_nP_r = \frac{n!}{(n-r)!}
\][/tex]
Here, [tex]\( n = 6 \)[/tex] and [tex]\( r = 6 \)[/tex]. Let's plug these values into the formula.
First, we need to calculate [tex]\( n! \)[/tex] which is [tex]\( 6! \)[/tex]:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
Next, we need to calculate [tex]\( (n-r)! \)[/tex]. In this case, [tex]\( n-r = 6-6 = 0 \)[/tex]. So, we need to find [tex]\( 0! \)[/tex]:
[tex]\[
0! = 1 \, \text{(by definition)}
\][/tex]
Now substitute these values back into the formula:
[tex]\[
{}_6P_6 = \frac{6!}{(6-6)!} = \frac{720}{1} = 720
\][/tex]
So, the value of [tex]\( {}_6P_6 \)[/tex] is:
[tex]\[
\boxed{720}
\][/tex]
