Given the question:
The manager of a baseball team has 15 players to choose from for his
9-person batting order. How many different ways can he arrange the
players in the lineup?
5,005
362,880
3,603,600
1,816,214,400
The number of was in which a smaller group is choosing from a bigger group is given by the bigger number combination the smaller number.
[tex]^nC_r= \frac{n!}{(n-r)!r!}[/tex]
Here, we want to choose a 9-person batting order from a team of 15 players. This is given by:
[tex]^{15}C_9= \frac{15!}{(15-9)!9!}=\frac{15\times14\times13\times12\times11\times10\times9!}{6!\times9!}= \frac{3,603,600}{720} =5,005[/tex]
