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?

- Pulsified333

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- Pulsified333

@jim_thompson5910 @satellite73 @dan815

- jim_thompson5910

`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

- Pulsified333

yes, i know the wording is odd

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## More answers

- jim_thompson5910

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

- jim_thompson5910

`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?

- Pulsified333

ok

- Pulsified333

56

- jim_thompson5910

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

- Pulsified333

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

- jim_thompson5910

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

- jim_thompson5910

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

- Pulsified333

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

- jim_thompson5910

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

- jim_thompson5910

Sorry I'm not really helpful with this one

- Pulsified333

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

- jim_thompson5910

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)

- Pulsified333

ok

- jim_thompson5910

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.

- Pulsified333

there is only 1 way if jackie gets the role

- jim_thompson5910

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

- Pulsified333

so what would the answer be

- Pulsified333

168?

- jim_thompson5910

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

- Pulsified333

6

- jim_thompson5910

jackie can't get the role though
6-1 = 5
so it's 5 actually

- Pulsified333

ah that makes sense

- jim_thompson5910

` 5 children audition`
`2 child roles available.`
how many ways to fill up these slots?

- Pulsified333

20

- jim_thompson5910

good

- Pulsified333

168*5*20 = 16800

- jim_thompson5910

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

- jim_thompson5910

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

- jim_thompson5910

that's just half the answer though

- Pulsified333

oh

- jim_thompson5910

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

- jim_thompson5910

give it a shot and tell me what you get

- Pulsified333

10080 if jackie gets the role

- Pulsified333

do we subtract them or add them together?

- jim_thompson5910

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

- jim_thompson5910

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

- Pulsified333

sorry 6720

- jim_thompson5910

better

- jim_thompson5910

we have 2 cases
George gets the role: 16800 different ways
Jackie gets the role: 6720 different ways
add them up

- Pulsified333

okay so that is 23520

- jim_thompson5910

yes

- jim_thompson5910

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

- Pulsified333

it was the correct answer

- Pulsified333

:D

- Pulsified333

thanks man

- jim_thompson5910

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

- jim_thompson5910

no problem

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