A director is casting the roles of Vladimir and Estragon in his play.

Question

A director is casting the roles of Vladimir and Estragon in his play. In all, a total of nine actors auditioned for at least one of the roles. Six auditioned for the role of Vladimir, and seven auditioned for the role of Estragon. If the director will choose one actor for each role from among those who auditioned for that role, and if the same actor cannot be chosen for both roles, in how many ways can the director cast the two roles?

A. 32
B. 36
C. 38
D. 42
E. 72

A. 32
B. 36
C. 38
D. 42
E. 72

GyMrAT · Accepted Answer

Total # of actors = 9
Actors auditioned for role of Vladimir = 6
Actors auditioned for role of Estragon = 7

Hence Actors auditioned for the role of both = (6 + 7) - 9 = 4

Therefore Actors who auditioned exclusively for the role of Vladimir = 6 - 4 = 2 & Actors who auditioned exclusively for the role of Estragon = 7 - 4 = 3

Now three cases are available for the director to choose the actors from:

Case 1 - For Vladimir, choose from actors who exclusively auditioned for Vladimir & for Estragon choose from actors who auditioned for both roles & exclusively for Estragon =  2C1 * 7C1  = 14

Case 2 - For Estragon, choose from actors who exclusively auditioned for Estragon & for Vladimir choose from actors who auditioned for both roles & exclusively for Vladimir =  3C1 * 6C1  = 18

Case 3 - Choose for both, from actors auditioned for both roles =  4C2  = 6

Total # of ways = 14 + 18 + 6 = 38.

Answer C.

Thanks,
GyM

Diwakar003 · Answer

Given total # of actors who attended the audition is 9 which is n(AuB). n(A) = 6, n(B)=7, n(AnB)=4. 
So 4 of them auditioned for both roles. 
Total ways of choosing is 6*7=42.
Out of these 42 ways there are 4 ways in which a same person can be selected for both casts. Hence 42-4=38.

Hence C.

Cheers!

mohshu · Answer

hi  Vyshak   Engr2012  any better way to solve this problem

JeffTargetTestPrep · Answer

Since 6 actors are auditioning for the role of Vladmir, there are 6P1 = 6!/(6-1)! = 6 ways that role can be filled.

Since a total of 9 actors auditioned for at least one of the two roles, 6 auditioned for the role of Vladimir, and 7 auditioned for the role of Estragon, it must be that 6 + 7 - 9 = 4 actors auditioned for both roles, 6 - 4 = 2 auditioned for Vladimir only, and 7 - 4 = 3 auditioned for Estragon only.

Let’s make two cases: 1) the director picks 1 of the 4 dual-role actors for Vladimir, and 2) the director picks 1 of the 2 single-role actors for Vladimir.

Case 1: 
If the director picks any 1 of 4 actors who auditioned for both roles to play Vladimir, then he has 6 actors left who can play Estragon (notice that 7 - 1 = 6 and we subtract 1 from 7 because that actor is chosen to play Vladimir already). Thus, the number of ways this can be done is 4 x 6 = 24.

Case 2: 
If the director picks any 1 of 2 actors who auditioned for Vladimir only to play Vladimir, then he has 7 actors to choose from to play Estragon. Thus, the number of ways this can be done is 2 x 7 = 14.

Thus, the total number of ways the director can choose the actors to play the two roles is 24 + 14 = 38.

Alternative solution:

We’ve approached the problem from the angle of Vladimir. We could have instead approached it from the angle of Estragon. Again, we divide it into two cases: 1) the director picks 1 of the 4 dual-role actors for Estragon, and 2) the director picks 1 of the 3 single-role actors for Estragon.

Case 1: 
If the director picks any 1 of 4 actors who auditioned for both roles to play Estragon, then he has 5 actors left who can play Vladimir (notice that 6 - 1 = 5 and we subtract 1 from 6 because that actor is chosen to play Estragon already). Thus, the number of ways this can be done is 4 x 5 = 20.

Case 2: 
If the director picks any 1 of 3 actors who auditioned for Estragon only to play Estragon, then he has 6 actors to choose from to play Vladimir. Thus, the number of ways this can be done is 3 x 6 = 18.

Thus, the total number of ways the director can choose the actors to play the two roles is 20 + 18 = 38.

Answer: C

Funsho84 · Answer

Another way to approach this problem is to assign letters to the 9 people who auditioned: ABCDEFGHI

Total ways = 7C1 * 6C1 = 42

Those who auditioned for V are ABCDEF
Those who auditioned for E are CDEFGHI

Now overlap the two:

ABCDEF....
 ...CDEFGHI

You will see that CDEF auditioned for both roles. So you can subtract 4 from the total.

42-4 = 38

Mohammadmo · Answer

9=6+7-x
x=13-9=4
7C1*6C1-4C2=42-6=36
Option C

Posted from my mobile device

globaldesi · Answer

I did it little differently; I got my answer correct somehow I am not sure if my cases taken are correct or not

Total # of actors = 9
Actors auditioned for role of Vladimir = 6
Actors auditioned for role of Estragon = 7

Hence Actors auditioned for the role of both = (6 + 7) - 9 = 4 . Let me name them starting from A to I

A B C D E F G H I 
where A B are only for Vladimir
and G H I are only for Estragon
and C D E F are for both
3 scenarios:
1 ) take 1 from A B C D E F for Vladimir and from G H I for Estragon
6*3 = 18
2 ) take 1 from C D E F G H I for Estragon and from A B for Vladimir
7*2= 14
3) take 1 for Vladimir from C D E F and 1 for Estragon from C D E F
4*3 = 12
But in 1 and 2 we have considered double scenarios example A H is considered in both
substract duplicates 3*2 = 6
so total = 18+14+12 -6 = 38

Bunuel  is this approach correct?

KarishmaB · Answer

6 - auditioned for V, 7 - auditioned for E
Total 9 auditioned for at least one.

9 = 6 + 7 - Both
Both = 4
So 4 people auditioned for both roles.

So 2 - ONLY V, 3 - ONLY E, 4 - BOTH

There are 2 cases:

V is picked from Only V category - 2 ways to pick V and 7 ways to pick E - Total 14 ways

V is picked from Both category - 4 ways to pick V and 6 ways to pick E - Total 24 ways

Total we have 14 + 24 = 38 ways.

Answer (C)

bumpbot · Answer

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A director is casting the roles of Vladimir and Estragon in his play.

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A director is casting the roles of Vladimir and Estragon in his play.

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