To evaluate the expression [tex]\({}_8 P_3\)[/tex], we need to calculate the number of permutations of 8 items taken 3 at a time. The formula for permutations is given by:
[tex]\[
{}_n P_r = \frac{n!}{(n-r)!}
\][/tex]
In this case, [tex]\(n = 8\)[/tex] and [tex]\(r = 3\)[/tex]. Let's apply the formula step-by-step:
1. Calculate [tex]\(n - r\)[/tex]:
[tex]\[
n - r = 8 - 3 = 5
\][/tex]
2. Evaluate [tex]\(n!\)[/tex]:
[tex]\[
n! = 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\][/tex]
3. Evaluate [tex]\((n - r)!\)[/tex]:
[tex]\[
(n - r)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\][/tex]
4. Divide [tex]\(n!\)[/tex] by [tex]\((n - r)!\)[/tex]:
[tex]\[
{}_8 P_3 = \frac{8!}{5!} = \frac{40320}{120} = 336.0
\][/tex]
Hence, the value of [tex]\({}_8 P_3\)[/tex] is [tex]\(336.0\)[/tex].
