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a) 6C2/12C4=1/33
b) (12C1x10C1x8C1x6C1)/(12C4x4!)=16/33
c) 1-a)-b)=1-17/33=16/33

OR (6C1x10C1x8C1)/(12C4x2!)=16/33
Sure there is easier way but this is the first solution came into my head
If you need more explanation, I'll give them later

a) 6C2/12C4=1/33 b) (12C1x10C1x8C1x6C1)/(12C4x4!)=16/33 c) 1-a)-b)=1-17/33=16/33

OR (6C1x10C1x8C1)/(12C4x2!)=16/33 Sure there is easier way but this is the first solution came into my head If you need more explanation, I'll give them later

So for 1), we treat each couple as one unit, and we need to pick 2 out of this six units. We get 6C2, and 12C4 means the number of combinations we can get from 12 people. Ok, this one I get it.

For 2), what is the logic to use 12C1 x 10C1 x 8C1 x 6C1 and 12C4 x 4!?

b) no married couple among 4 people
first person you can choose in 12C1 ways (any of 12 people)
second person 10C1 (any of left but not the spouse of the first chosen 11-1=10)
third the same logic (neither the spouse of the first nor the second's) 8C1
fourth 6C1
But to exclude the same fourths with different order we divide it by 4!
To get the probability we should divide it by the number of all possible fourths 12C4

tried to do my best. Did it help?
What are the answers?

It is tough for me too. I agree with the answers given by lovely_baby.

(a) 6C2/12C4 = 1/33
(b)The number of ways the chosen 4 can be unmarried = 12C1 * 10 C1 * 8C1 * 6C1 = 12*10*8*6

Because the above number is a permutation (order is important), we need to convert it into combination (order is not important as we are choosing) i.e., The actual ways without regard for order = (12*10*8*6)/4!
= 5*8*6
The required probability = (5*8*6)/12C4 = 16/33

(c) The probability that exactly one is married couple = 1-(a)-(b) = 16/33

Can anybody solve (c) in the way we have solved (a) & (b)? Somehow, I am in the loop and unable to come out of it.

neelesh wrote:

This question is probably a simple one for most people on this forum here, but wanted to see what approach to use for such questions. Somehow my appeoach is not matching with the OA.

Six Married couples are standing in a room. If 4 people are chosen at random, find the probability p for the following scenarios

a) 2 married couples are chosen b) no married couple is among the 4 c) exactly one married couple is among the 4.

Especially for mallelac!!!
From my previous message
c) (6C1x10C1x8C1)/(12C4x2!)=16/33
6C1 one of 6 couples
10C1 one of left 10 people
8C1 one more of left 8
divide by 2! because doesn't matter the order

We need to divide by 3! because the number of ways with 3 items (1 couple and 2 members who are not a couple) are calculated here and order is not important here. infact I too got the answer with 2! division but could not justify it.

lovely_baby wrote:

Especially for mallelac!!! From my previous message c) (6C1x10C1x8C1)/(12C4x2!)=16/33 6C1 one of 6 couples 10C1 one of left 10 people 8C1 one more of left 8 divide by 2! because doesn't matter the order

I don't know how to explain better in English, but in this case the couple doesn't matter. You just choose one and forget about it . But while choosing the singles you twice them...mmm... is it a bit more clear?

It is tough for me too. I agree with the answers given by lovely_baby.

(a) 6C2/12C4 = 1/33
(b) While choosing 4, no couple shoule be present. Let us pick 1 out of 12. In the remaining 12, the first person's spouse is present and thus has to be skiiped because no couple should be present in 4. So, we can choose 2nd person out of 10. This 2nd person's spouse is present in the remaining 9. So, the 3rd person has to be from the remaining 8. Similarly, 4th person has to be from the remaining 6.

The total number of ways the chosen 4 can form no couples = 12*10*8*6
Here, all the number of ways are accounted i.e. ABCD, ABDC, ADCB etc. But, when we pick 4 persons at random, they can be in any order. All of ABCD, ABDC, ADCB etc are one and the same as for as the group is concerned. Thus, we need to divide the (12*10*8*6) with 4!

In other words,
Because the above number is a permutation (order is important), we need to convert it into combination (order is not important as we are choosing) i.e., The actual ways without regard for order = (12*10*8*6)/4!
= 5*8*6
The required probability = (5*8*6)/12C4 = 16/33

(c) If you observe care fully, 1 = P(a) + P(b) + P(c) from the set theory because Total probability (i.e., 1) = Probability that two married couples are chosen + Probability that one married couple is chosen + Probability that NO married couple is chosen

The probability that exactly one is married couple = 1-P(a)-P(b) = 16/33

The above method is easy by observation. However, if P(c) is asked straight, then you may work out as follows.

Let P(c) = Probability that exactly one is married

The number of ways of choosing 4 such that only 1 is married couple = 6C1 * 10C2 - 6C1*5C1 = 6*45 - 6 * 5 = 240

This is very well explained d-dogg

"6C1 ways to select one couple (A A)

10C2 ways to select the remining two people (B C) - but this includes selecting another couple (like B B, or C C); so subtract 5C1

Favorable combinations = 6C1*(10C2 - 5C1) = 240 "

Total space = 12C4

P(c) = 240/12C4 = 16/33

The second method is easier if you understand the logic in it. It is faster too. If you are uncomfortable with it, go for the first method where you should be good at breaking up the unknown into known problems.

neelesh wrote:
This question is probably a simple one for most people on this forum here, but wanted to see what approach to use for such questions. Somehow my appeoach is not matching with the OA.

Six Married couples are standing in a room. If 4 people are chosen at random, find the probability p for the following scenarios

a) 2 married couples are chosen
b) no married couple is among the 4
c) exactly one married couple is among the 4.

christoph wrote:

HongHu wrote:

ssumitsh wrote:

(c) 8 * 6C1 * 5C3

What's wrong with this?

can you plz explain the second and the third answer ? what would be the best concept to solve them ? thx

This is a very good question. Permutation and combination questions often have multiple solutions that could reach the same answer. Allow yourself to think more than just gets the answer will give yourself an opportunity to really understand the concept and master the skills to solve this kind of questions. That is why I would like to nudge people to think more.

Love_baby has a good approach and he had explained it clearly. Ssumitsh also has a good approach. He only answered the number of outcomes, but it can be easily converted to probabilities. However, his answer for (c) is different from the right answer. Why?

700Plus wrote:

I dont understand 8*6C1*5C3? Can you explain how it came ?

To understand his logic, I would explain his solution to (b) and see if anybody can understand his logic for (c) and found why he gave a wrong answer.

ssumitsh wrote:

(b) 16 * 6C4

(b) asks for probability that non of the four people are couples. So you first pick four couples out of the six couples. C(6,4) Then you get one person from each of the four picked couples. For each couple you have two choices. (C(2,1)=2)
Therefore the combined outcome is: C(6,4)*2*2*2*2=240

Now we know that (c) is wrong. But his line of thought is right. So what did he do wrong?

(a) Choose the first person in 12 ways, the next person chosen in 1 way (his partner), the 3rd person in 10 ways and then the last person (his partner) in one way so 10*12 = 120 ways

(b) 12 ways * 10 ways * 8 ways * 6 ways (skipping one partner for each)
= 5760