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# Given that there are 6 married couples. If we select only 4

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Given that there are 6 married couples. If we select only 4 [#permalink]

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18 Jan 2008, 01:05
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Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?
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18 Jan 2008, 01:33
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$$p=\frac{12}{12}*\frac{10}{11}*\frac{8}{10}*\frac{6}{9}=\frac{16}{33}$$

or

$$p=\frac{C^6_4*(C^2_1)^4}{C^{12}_4}=\frac{16}{33}$$

or

$$p=\frac{P^6_4*(P^2_1)^4}{P^{12}_4}=\frac{16}{33}$$
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Last edited by walker on 19 Jan 2008, 02:53, edited 3 times in total.
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18 Jan 2008, 16:03
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dominion wrote:
Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?

Prob: 1*10/11*8/10*6/9 = 16/33
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19 Jan 2008, 02:41
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dominion wrote:
walker wrote:
$$p=\frac{C^6_4*(C^2_1)^4}{C^{12}_4}=\frac{16}{33}$$

$$C^6_4$$ - we choose 4 couples of 6 ones.

$$C^2_1$$ - we chose one people of 2 ones for one couple.

$$(C^2_1)^4$$ - we have 4 couple and for each we choose one people of 2 ones for one couple.

$$C^{12}_4$$ - the total number of combinations to choose 4 people from 12 people.
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Last edited by walker on 19 Jan 2008, 02:43, edited 1 time in total.
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19 Jan 2008, 02:52
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$$p=\frac{12}{12}*\frac{10}{11}*\frac{8}{10}*\frac{6}{9}=\frac{16}{33}$$

$$\frac{12}{12}$$ - we choose 12 of 12.

$$\frac{10}{11}$$ - we choose 10=12-1(prevous choice)-1(another people out of couple) of 11=12-1(prevous choice).

$$\frac{8}{10}$$ - we choose 8=12-2(prevous choice)-2(another people out of couple) of 10=12-2(prevous choice).

$$\frac{6}{9}$$ - we choose 6=12-3(prevous choice)-3(another people out of couple) of 9=12-3(prevous choice).

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$$p=\frac{P^6_4*(P^2_1)^4}{P^{12}_4}=\frac{16}{33}$$
the same logic as for $$C_m^n$$
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24 Aug 2008, 14:46
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dominion wrote:
Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?

= (12C1 * 10C1 * 8C1*6C1)/4!/ 12C4
= 12*10*8*6 / (12*11*10*9) = 16/33
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05 Feb 2009, 18:36
Expert's post
$$C^6_4$$ - we choose 4 couples (not people) out of 6 couples (the number of all couples)

the next step: we choose one person out of each couple - $$C^2_1$$
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05 Feb 2009, 20:00
walker wrote:
$$C^6_4$$ - we choose 4 couples (not people) out of 6 couples (the number of all couples)

the next step: we choose one person out of each couple - $$C^2_1$$

Walker,
Ok, its making better sense now.
Are we choosing 4 couples because that means "not 4 people", ie 12-4=8?
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06 Feb 2009, 00:50
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dominion wrote:
Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?

So, we should count how many possibilities we have to form group of 4 people with restriction: none of them are married to each other.
There a few ways to count all combinations. One of the ways is following: 4 people belong to 4 distinct couples. Therefore, we could choose these 4 couples and then 1 person out of each couple - $$C^6_4*(C^2_1)^4$$

look at other problem:
Quote:
Given that there are 8 soccer teams. If we select only 6 people out of the 88 (8 teams, 11 people in each team), what is the probability that none of them are out of the same team?

we can use the same reasoning: choose 6 teams out of 8 teams (our 6 people are from 6 different teams) and then choose 1 player out of 11 for each team.

I hope it is clearer now.
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06 Feb 2009, 10:22
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dominion wrote:
Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?

total ways of choosing 4 out of 12:12x11x10x9
first person can be chosen in 12 ways. The next person can be chosen in 10 ways (because we don't want spouses to be in the group). The next person can be again chosen in only 8 ways (out of the 10 people left, we have to exclude 2 whose spouses we have already selected). And the last person in 6 ways.

hence, probability=12x10x8x6/(12x11x10x9) = 16/33.
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06 Feb 2009, 19:06
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botirvoy wrote:
x2suresh wrote:
dominion wrote:
Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?

= (12C1 * 10C1 * 8C1*6C1)/4!/ 12C4
= 12*10*8*6 / (12*11*10*9) = 16/33

suresh, can you please explain the logic of this calculation?
Thank you!

12C1 = select any person from 6 married couples (12 person)
10C1 = select second person from remaining people and exclude the first person's spouse
8C1 = select 3rd person from remain people exclue first and second perssons's spouses
6C1 = select 4th person from remaining people exclude 1st,2nd ,3rd persons's spouse

Becuase order is not matter.. you need to divide by 4!

Did you get it?

=
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23 Mar 2009, 11:20
xALIx wrote:
walker wrote:
dominion wrote:
Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?

So, we should count how many possibilities we have to form group of 4 people with restriction: none of them are married to each other.
There a few ways to count all combinations. One of the ways is following: 4 people belong to 4 distinct couples. Therefore, we could choose these 4 couples and then 1 person out of each couple - $$C^6_4*(C^2_1)^4$$

look at other problem:
Quote:
Given that there are 8 soccer teams. If we select only 6 people out of the 88 (8 teams, 11 people in each team), what is the probability that none of them are out of the same team?

we can use the same reasoning: choose 6 teams out of 8 teams (our 6 people are from 6 different teams) and then choose 1 player out of 11 for each team.

I hope it is clearer now.

Wow, makes very clear sense now. Much obliged.
Ali

another way of looking at this problem :

total number of ways = 12C4 = 495

now let's try to find out total number of unfavourable ways. We can subtract those from 495 to get the number of favourable ways.

number of ways of choosing 4 people such that there are two couples = 6C2 = 15

number of ways of choosing 4 people such that there is only one couple = number of ways of choosing one couple * number of ways of choosing two people who are not couples = 6C1 (10C2 -5) = 240.

total number of unfavourable ways = 240 + 15 = 255

total number of favourable ways = 495 - 255 = 240

required probability = 240/495 = 16/33
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27 Sep 2009, 06:13
Given that there are 6 married couples. If we select only 4 people out of the 12, what is the probability that none of them are married to each other?

Soln:
= (12/12) * (10/11) * (8/10) * (6/9)
= 16/33
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31 Mar 2010, 13:17
hi walker regarding the soccer question the answer should be

8C6 * (11C1)^6
---------------
88C4

right na thks

but just want to know if i have understood the concept thks
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03 Apr 2010, 06:17
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bnagdev282 wrote:
hi walker regarding the soccer question the answer should be

8C6 * (11C1)^6
---------------
88C4

right na thks

but just want to know if i have understood the concept thks

Sorry, where did you get 88C4? It's a huge number!
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03 Apr 2010, 15:22
hi walker i calculated it as 8 team with 11 players each so in all 88 players and chosing 4 out of them so 88c4
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03 Apr 2010, 15:46
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bnagdev282 wrote:
hi walker i calculated it as 8 team with 11 players each so in all 88 players and chosing 4 out of them so 88c4

I see... You are right if we use 88C6 rather than 88C4 (we need 6 people). I think it is a typo, right?
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03 Apr 2010, 15:48
ya typo error sorry

thks walker
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11 Apr 2011, 07:43
In x2suresh's solution, I'm unable to visualize the exact arrangements because of which we should divide 12x10x8x6 ways by 4!. Could someone please explain in more detail ? The thing is, we're choosing here, so how does the question of arrangements come in picture at all ?

Regards,
Subhash
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30 Apr 2011, 20:14
1 * 10/11 * 8/10 * 6/9
= 16/33
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Re: combinatorics   [#permalink] 30 Apr 2011, 20:14

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