Two canoe riders must be selected from each of two groups of campers.

Question

Two canoe riders must be selected from each of two groups of campers. One group consists of three men and one woman, and the other group consists of two women and one man. What is the probability that two men and two women will be selected?

(A) 1/6
(B) 1/4
(C) 2/7
(D) 1/3
(E) 1/2

(A) 1/6
(B) 1/4
(C) 2/7
(D) 1/3
(E) 1/2

Shadyshades · Answer

Conditions-

I have to take two members from each group. 
AND
 I have to take 2 Men and 2 Women in total.

Group A has 3 Men and 1 Woman
Group B has 1 Man and 2 Women

Case 1  - I can choose 2 Men from Group A and 2 Women from Group B

so the number of combinations are 3C2*2C2 = 3*1= 3

Case 2  - I can choose 1 Man 1 Woman from Group A and 1 Man 1 Woman from Group B

so the number of combinations are 3C1*1C1*1C1*2C1 = 3*1*1*2= 6

Total number of combinations are 4C2*3C2= 6*3= 18

(Case 1 + Case 2) / Total combinations

=(6+3)/18
=9/18
=1/2 
Option E

unraveled · Answer

The crux of the question is that two riders must be selected from each of the two groups. Had this not been the case, the four riders can be selected as usual in  7_{C_4}  ways.
Required selection can be done in ways:
1. For 1st group(3M,1W) -  4_{C_2}  
2. For 2nd group(2W,1M) -  3_{C_2} 
Hence,  \frac{4*3}{2}*\frac{3*2}{2} = 18

Now, 2M and 2W can be selected in ways:
A) 2M from 1st group and 2W from 2nd group -  3_{C_2}*2_{C_2} = 3 
B) 1M and 1W from each group -  3_{C_1}*1_{C_1}*2_{C_1}*1_{C_1} = 6

P(2M,2W) =  \frac{3+6}{18} = \frac{1}{2}

Answer E.

Wadree · Answer

First team = man1, man2, man3, woman  
Possible pick of 2 =  ,  ,  ,  ,   and  
So.... total scenario = 6 ...... scenario of 2 men = 3 .....so probability of 2 men = 3 / 6 = 1 / 2 .......
Scenario of 1 woman & 1 man = 3 ...... so probability of 1 woman & 1 man = 3 / 6 = 1 / 2 ......

Second team = woman1, woman2, man  
Possible pick of 2 =  ,  ,   
So... total scenario = 3 ... scenario of 1 man & 1 woman = 2 ....so probability of 1 man & 1 woman = 2 / 3 .....
Scenario of 2 women = 1 ...... so probability of 2 women = 1 / 3 .......

Now....2 man & 2 woman can be taken in 2 ways......

2 W from second team and 2 M from first team...probability of dis scenario =   ×   = 1 / 6

1 M & 1 W from 2nd team and 1 M & 1 W from 1st team...probability of dis scenario =   ×   = 2 / 6

So....all probability =   +   = 3 / 6 = 1 / 2

! nah id win!

Kinshook · Answer

Two canoe riders must be selected from each of two groups of campers. One group consists of three men and one woman, and the other group consists of two women and one man.

What is the probability that two men and two women will be selected?

First group: 3M + 1W
Number of ways to select 2 canoe riders = 4C2 = 6

Second group: 2W + 1M
Number of ways to select 2 canoe riders = 3C2 = 3

Total number of ways to select 2 canoe riders = 6*3 = 18

Ways to select 2M + 2W: - 
1M + 1W from first group & 1W + 1M from second group; Number of ways = 3C1*1 *2C1*1 = 3*2= 6
2M from first group and 2W from second group; Number of ways = 3C2*2C2 = 3
Total number of ways to select 2M + 2W = 6+3 = 9

The probability that two men and two women will be selected = 9/18 = 1/2

IMO E

CelestialBro · Answer

there are 2 ways in which 2M and 2W can be chosen from the 2 given groups:

case 1: 
P(MM) from G1= 3/4 * 2/3 * 2!/2! = 1/2 
P(WW) from G2= 2/3 * 1/2 *2!/2! = 1/3
P(MM) and P(WW)= 1/2 * 1/3 = 1/6

case 2:
P(MW) from G1 = 3/4 * 1/3 * 2! = 1/2
P(MW) from G2 = 1/3 * 2/2 * 2! = 2/3
P(MW) and P(MW) = 1/2 * 2/3 = 1/3

So total probability can be calculated by combining the probabilities from the above two cases:
case 1 or case 2 = 1/6 + 1/3 = 9/18 = 1/2

HarshavardhanR · Answer

Group 1:  3M, 1W
 Group 2 : 1M, 2W

Two riders to be chosen from each group.

To get two men and two women, what are the possible cases?

(1) 2 Men from Group 1 AND 2 Women from Group 2.

3C2 x 2C2 = 3.

(2) 1 Man and 1 Woman from Group 1 AND 1 Man and 1 Woman from Group 2

3C1 x 1C1 x 1C1 x 2C1 = 3 x 1 x 1 x 2 = 6

No other cases are possible.

Total Outcomes = Number of ways of choosing 2 from Group 1 AND 2 from Group 2 = 4C2 x 3C2 = 6 x 3 = 18.

Our required Probability = (3 + 6) / 18 = 9/18 =  1/2. Choice E .

---
Harsha

Two canoe riders must be selected from each of two groups of campers.

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Two canoe riders must be selected from each of two groups of campers.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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Perfect 805 Score Debrief by Julia

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