Question

For this problem, assume 9 males audition, one of them being George, 6 females audition, one of them being Jackie, and 5 children audition. The casting director has 3 male roles available, 1 female role available, and 2 child roles available. How many different ways can these roles be filled if exactly one of George and Jackie gets a part?

Answer 1

@jim_thompson5910 @satellite73 @dan815

Answer 2

`if exactly one of George and Jackie gets a part`

so they mean just one of them? either George OR Jackie? The wording they have is a bit odd

Answer 3

yes, i know the wording is odd

Answer 4

I'm going to assume they meant to say George OR Jackie (exactly one person) gets the role. So it's not possible for both to get the role

Answer 5

`9 males audition`
`3 male roles available`

Let's say George is locked in to get a role. So we really have 3-1 = 2 slots left for the male roles. This is from a pool of 9-1 = 8 males

how many ways are there to fill 2 slots from a pool of 8 males?

Answer 6

ok

Answer 7

56

Answer 8

order doesn't matter. So you divide 56 by 2 to get 28

Answer 9

okay but isnt it possible to have any of the three roles

Answer 10

I guess it depends on what the roles are. Are they important roles or extras? Hmm I'm not sure

Answer 11

if the roles are important (eg: leading roles), then order would matter since the characters are different

Answer 12

so then how do we account for the three roles he could play

Answer 13

I'm not sure now because I'm not sure what the roles are. So I don't know if order matters or not

Answer 14

Sorry I'm not really helpful with this one

Answer 15

well i have 4 attempts so lets do it with three separate roles on this

Answer 16

ok well if order did matter, then George could have male role A, male role B, male role C

he has 3 choices

if he picks male role A, then there are 8*7 = 56 ways to fill up the other 2 male roles
if he picks male role B, then there are 8*7 = 56 ways to fill up the other 2 male roles
if he picks male role C, then there are 8*7 = 56 ways to fill up the other 2 male roles

as a shortcut, there are 3*56 = 168 ways to fill up the male roles assuming George gets a male role and order matters (the roles are important roles)

Answer 17

ok

Answer 18

how many ways are there to fill the female role? We're still under the assumption George got the role. So Jackie can't get the role if George did.

Answer 19

there is only 1 way if jackie gets the role

Answer 20

like I said ` We're still under the assumption George got the role. So Jackie can't get the role if George did.`

Answer 21

so what would the answer be

Answer 22

168?

Answer 23

we're not there yet

how many ways are there to fill the female role? assume that Jackie can't get the role

Answer 24

6

Answer 25

jackie can't get the role though

6-1 = 5

so it's 5 actually

Answer 26

ah that makes sense

Answer 27

` 5 children audition`
`2 child roles available.`

how many ways to fill up these slots?

Answer 28

20

Answer 29

good

Answer 30

168*5*20 = 16800

Answer 31

summary so far

IF george gets a role, then there are...

168 ways to fill up the male roles
5 ways to fill up the female role
20 ways to fill up the child roles

Answer 32

yes you beat me to it, you multiply those values out

Answer 33

that's just half the answer though

Answer 34

oh

Answer 35

we haven't considered the scenario that Jackie gets the role

Answer 36

give it a shot and tell me what you get

Answer 37

10080 if jackie gets the role

Answer 38

do we subtract them or add them together?

Answer 39

once again, if either Jackie or George gets the role, then the other person can't get it

so we're now assuming Jackie gets the role. So george did NOT get the role. Try again

Answer 40

if george did NOT get the role, then the pool of 9 males drops to 8

Answer 41

sorry 6720

Answer 42

better

Answer 43

we have 2 cases

George gets the role: 16800 different ways
Jackie gets the role: 6720 different ways

add them up

Answer 44

okay so that is 23520

Answer 45

yes

Answer 46

which is the final answer. It's the number of ways to pick all of the cast based on those specific conditions

Answer 47

it was the correct answer

Answer 48

:D

Answer 49

thanks man

Answer 50

ok so it was a lucky guess on our part to assume order mattered

Answer 51

no problem