Question

A baseball coach needs to choose 99 players to be in the batting lineup for the next game. There are 44 freshmen, 44 sophomores, 55 juniors, and 44 seniors on the team. Step 1 of 2: How many ways can the batting order be chosen if the coach wants no more than 22 juniors to be in the lineup? Express your answer in scientific notation rounding to the hundredths place.

Answer 1

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 = $\binom{12}{9}$ = 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

  = $\binom{12}{8} \times \binom{5}{1} = 2475$

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

   = $\binom{12}{7} \times \binom{5}{2} = 7920$

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.