Answer: The required number of ways is 335221286400.
Step-by-step explanation: Given that nineteen students play on a baseball team.
We are to calculate the number of ways in which the batting order of 10 players be arranged.
Since it is a situation of arrangement, the total number of ways will be equal to the permutation of 19 taken 10 at a time.
The formula for the number of permutations of n things taken r ata time is given by
[tex]^nP_r=\dfrac{n!}{(n-r)!}.[/tex]
From the given information, we have
n = 19 and r = 10.
Therefore, the number of arrangements is given by
[tex]^{19}C_{10}\\\\\\=\dfrac{19!}{(19-10)!}\\\\\\=\dfrac{19\times18\times17\times16\times15\times14\times13\times12\times11\times10\times9!}{9!}\\\\\\=19\times18\times17\times16\times15\times14\times13\times12\times11\times10\\\\=335221286400.[/tex]
Thus, the required number of ways is 335221286400.
