Answer:
There are 18,162,144,000 possible starting line-ups
Step-by-step explanation:
The order in which the player are chosen is important, as if two player exchange places, we have a new batting order. So we use the permutations formula to solve this question.
Permutations formula:
The number of possible permutations of x elements from a set of n elements is given by the following formula:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
In this question:
Batting order:
9 batters(8 field + the designated hitter) from a set of 15. So
[tex]B = P_{(15,9)} = \frac{15!}{(15-9)!} = 1816214400[/tex]
Pitcher:
1 from a set of 10. So
[tex]P = P_{(10,1)} = \frac{10!}{(10-1)!} = 10[/tex]
Total:
[tex]T = B*P = 1816214400*10 = 18162144000[/tex]
There are 18,162,144,000 possible starting line-ups
