A director is deciding how to cast a play from a pool of 50

Question

A director is deciding how to cast a play from a pool of 50 available actors and actresses. There are m male roles and n female roles. If each male actor can play any male part, and each female actress can play any female part, how many ways are there to cast the play?

(1) 60% of the acting pool is female.

(2) There are 116,280 ways to cast the male parts.

(1) 60% of the acting pool is female.

(2) There are 116,280 ways to cast the male parts.

smyarga · Accepted Answer

To answer the question we should know the total number of females and males available in the pool, n and m.
The solution for this is  P_f^n*P_{50-f}^m  if we consider all roles are different (otherwise  C_f^n*C_{50-f}^m ), where  f  is the total number of females.

Anyway, we have three unknown variables: n, m, f

(1) Insufficient. Give us only  f=0.6*50 . We still don't know  n, m

(2) Insufficient.  P_{50-f}^m=116,280 . One equation in two variables and we still don't know  n .

(1)+(2). We can find  m, f , but we don't have any info about  n . Insufficient.

The answer is E.

thelosthippie · Answer

The total number of ways in which the play can be cast is m*n.

The goal is to find m and n.

(1) 60% of the acting pool is female.
Total=50 hence females = 30 and males=20 
 -- But no clue about m,n

(2) There are 116,280 ways to cast the male parts.
 -- Alone this statement gives no idea about m,n or the split of male/female in the total fifty

Together:
we know there are 20 males and 116,280 ways to cast them hence can find out 'm'. But 'n' still remains unknown.

Please let me know if the above explanation is clear.

smyarga · Answer

Sorry, do you understand correctly the meaning "cast the play"? it means that for every role we should determine the actor or actress. Am I correct?

thelosthippie · Answer

Yeah I get it , the total number of ways to cast the play will not be m*n. I get your explanation. thanks

stne · Answer

What is the exact way to cast a play ? 
Is it not number of ways to select men (for the roles) from among the total men available * number of ways to select females (for the roles) from the total number of females available?

if it could be explained in words please first, then in formula it would be easier to understand

I thought  \hspace{8mm}C^{mr}_{males} * \hspace{8mm}C^{fr}_{females} \hspace{8mm}  where  mr  is the number of male roles and  fr is the number of female roles

Now assuming order is not important

(1)  60% of the acting pool is female : We get only males and females no information about how many male roles and how many female roles insufficient.

(2)  There are 116,280 ways to cast the male parts . No information about how many males or females or how many male roles or female roles

1+2 
we can find total men and men roles , we can find total females but cannot get total female roles hence insufficient.

I hope my interpretation is correct for the total number of ways to cast a play, please do correct if it is wrong.
Thanks

jlgdr · Answer

So you basically have x males 50 - x females and 'm' males roles and 'f' male roles How many arrangements? (1) Ok, so we know how many males and females but we don't have any info about number of roles Insuff (2) OK, so we can actually at best get the number of males and therefore imply number of females as well, and we can get the number of 'm' roles But we have no info on number of female roles (1) + (2) together Still not info about number of females roles Hence E is the correct answer Hope it helps Cheers! J

Crytiocanalyst · Answer

(1) 60% of the acting pool is female.
The above gives us information about the the distribution of the availability of the cast members however doesn't give us information in how the male and female can be cast there combination we cannot find with the given data

(2) There are 116,280 ways to cast the male parts.
This individually cannot answer how many male participants can be chosen from even if we use 50-f formulae

Even when combined we still will not be able to come to a conclusion since the variable and equations won't match 
Therefore IMO E

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A director is deciding how to cast a play from a pool of 50

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A director is deciding how to cast a play from a pool of 50

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