Chapter 10 - Section 10.2 - Permutations and Combinations - 10.2 Assess Your Understanding - Page 696: 58

See below.

If we want to choose $k$ elements out of $n$ regarding the order, not allowing repetition, we can do this in $_{n}P_k=\frac{n!}{(n-k)!}$ ways. The order matters here when choosing the eight batters (since the pitcher bats ninth), thus we have to use permutations. We have $8$ people for $8$ spots, thus $_{8}P_{8}=\frac{8!}{(8-8)!}8!=40320$