Answer :
Answer:
The number of ways the batting line up can be made is 3,851,971,200 ways.
Step-by-step explanation:
The coach of the baseball team needs to choose 9 players for the batting lineup.
He can choose from a total of 4 freshmen, 4 sophomores, 5 juniors and 4 seniors for the team.
He has to choose the team so that there at most 2 juniors on the lineup.
The total number of available player who are not juniors is 4 + 4 + 4 = 12
i) If the coach selects no juniors then there are 9 positions to fill up and
since the batting order is not specific so these 9 players can be selected
from 12 players = [tex]\binom{12}{9}[/tex] = 220 ways
ii) If the coach selects one junior then there are other 8 other positions to
fill up from 12 players and one junior is to be selected from 5 juniors
= [tex]\binom{12}{8} \times \binom{5}{1} = 2475[/tex]
iii) If the coach selects two juniors then there are other 7 other positions to
fill up from 12 players and two juniors are to be selected from 5 juniors
= [tex]\binom{12}{7} \times \binom{5}{2} = 7920[/tex]
So the ways of selecting the players is = 220 + 2475 + 7920 = 10615 ways
Since there is no specific order to the batting line up the number of ways the batting line up can be made = 9! × 10615 = 3,851,971,200 ways.