To evaluate the expression \({ }_8 P_8\), we need to use the formula for permutations, which is given by:
[tex]\[
{ }_n P_r = \frac{n!}{(n-r)!}
\][/tex]
Here, \( n = 8 \) and \( r = 8 \). Plugging these values into the formula, we get:
[tex]\[
{ }_8 P_8 = \frac{8!}{(8-8)!}
\][/tex]
Simplifying the expression in the denominator:
[tex]\[
{ }_8 P_8 = \frac{8!}{0!}
\][/tex]
We know that \(0!\) (zero factorial) is equal to 1. Therefore, the expression simplifies to:
[tex]\[
{ }_8 P_8 = \frac{8!}{1}
\][/tex]
[tex]\[
{ }_8 P_8 = 8!
\][/tex]
Now, we need to determine the value of \(8!\).
[tex]\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\][/tex]
Therefore,
[tex]\[
{ }_8 P_8 = 40320
\][/tex]
So, the simplified answer is:
[tex]\[
40320
\][/tex]
