As its star player, Alina becomes the captain of the Applied Math soccer team an

Question

As its star player, Alina becomes the captain of the Applied Math soccer team and is tasked with choosing the 10 other players from a set of 11 men and 7 women. a) In how many ways can Alina select the 10 other players? 43758 b) Suppose now that, of the 10 remaining players, there must be exactly 6 men and 4 women. In how many ways can Alina select the 10 other players? 16170 c) Suppose that, in addition to the restrictions posed in part b), Cody (one of the men Alina must select from) refuses to join the team. Now in how many ways can Alina choose the 10 remaining players? 7350 Suppose that, in addition to the restrictions posed in parts b) and c), Tyler (one of the men Alina must select from) will join the team if and only if Sam (another one of the men Alina must select from) joins. Now in how many ways can Alina choose the 10 remaining players? e) Suppose that, in addition to the restrictions posed in part b), Matt (one of the men Alina must select from) and Jenn (one of the women Alina must select from) refuse to play for the same team. In how many ways can Alina choose the 10 remaining players?

Explanation / Answer

11 men and 7 women for 10 players

a.

So this choosing 10 people from 7+10 = 17 players
So, that can happen in 17C10 ways = 19448 ways

b.

So, selecting 6 from 11 men and 4 from 7 women.
This can happen in 11C6 *7C4 = 16170

c.
You have to select 6 from 10 available mean and 4 from 7 women
So, 10C6*7C4 = 7350

d. Thats:

Case1 : Sam joins:

1 joins for sure(Sam joins) , in 1 way
Rest 9 are selected as 4 from men and 4 from women in:

1*10C5*7C4= 8820 ways

Case2: Sam doesnt join , then typer also won't join
So, from 9 remaining men and 7 women choose 6 men and 4 women in: 9C6*7C4
= 2940 ways

Total ways =11760 ways

e.

matt plays then ( and jena doesn't):

6 men from 10 remaining
4 women from 7 remaining

This can happen in 10C6*7C4 = 7350

Jena plays then ( and matt doen't):

7 men from 11 remaining
3 women from 6 remaining

This can happen in 11C7*6C3 = 6630

So , 13950 is total ways

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