A group of twelve volunteers, including Ariel and Naja, are helping to

Question

A group of twelve volunteers, including Ariel and Naja, are helping to organize a charity event. A committee consisting of a chair, vice-chair, secretary, and treasurer is made up of four volunteers randomly chosen from the twelve. What is the probability that Ariel is chosen as the chair of the committee, and Naja is also on the committee?

A. 1/11
B. 1/22
C. 1/44
D. 1/66
E. 1/132

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tgsankar10 · Answer

No of ways the committee of  4  can be formed out of  12 = 12*11*10*9

Since Ariel is chosen as the Chair, there are 3 positions remaining. So there are 3 ways where Ariel is the Chair and Naja is in the committee. Remaining 2 positions can be formed by choosing 2 out of remaining 10 volunteers in  10*9  ways

No of ways Ariel is the Chair and Naja in the committee  =3*10*9

Probability of desired event  = \frac{3*10*9}{12*11*10*9}=\frac{1}{44}

Answer: C

ManifestDreamMBA · Answer

Probability that Ariel is chosen as the chair of the committee = 1/12

Probability that Naja is also on the committee = 3/11

Probability of both events = 1/12 * 3/11 = 1/44

Missinga · Answer

Ariel can be chosen as the chair of the committee = 1/12 
 Naja can also be on the committee = 3/11
Required Probability= (1/12)*(3/11) = 1/44

Answer: C

Abhiswarup · Answer

Ariel can be chosen as the chair of the committee = 1C1/12
Naja can also be on the committee = 3C1/11 (12-1) 
Re Probability= (1/12)*(3C1/11) = 1/44

HarshavardhanR · Answer

Probability = #Favorable Outcomes / #Total Outcomes

#Total Outcomes = 12C4 (ways to select 4 people for the committee) x 4! (ways to fix who among the 4 takes which role) =  12C4 x 4!

#Favorable Outcomes --->

(1) 1C1 = 1 way to select Ariel as chair. She needs to be chosen as the chair. 
(2) 3C1 = 3 ways to bring Naja into the committee --- she could be vice-chair, or secretary, or treasurer. 
(3) 10C2 ways to choose the other 2 committee members from among the remaining 10 volunteers.
(4) 2! ways to fix which of the above two will take which role. One being treasurer and the other being secretary is different from the former being secretary and the latter being treasurer. Hence, 2!.

# Favorable Outcomes =  1 x 3 x 10C2 x 2

Required Probability = (1 x 3 x 10C2 x 2) / (12C4 x 4!) =  1/44.   Choice C .

---
Harsha

Adit_ · Answer

Order matters since Ariel is the chair of the committee.

Choosing remaining 10 for 2 seats:  
10C2 = 45.
Arranging Naja and the two chosen in:
3! ways = 6.

Thus probability = 45*6/12*11*10*9 =  1/44 .

A group of twelve volunteers, including Ariel and Naja, are helping to

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