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A group of 10 people consists of 3 married couples and 4 [#permalink]
 17 May 2011, 08:18
Question Stats:
0% (00:00) correct
100% (00:42) wrong based on 0 sessions
A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be from the 10 people. How many different committees can be formed if the committee can consist of at most 1 married couple? Can someone employ the forward method i.e. (not the 1-x method to deduce the answer) ? I think I over counting as my answer overshoots the OA Answer = No of ways to select one couple and two singles + No of ways to select all singles OA is 207.
Using 1 - x method the question is piece of cake. But if you employ the forward method i.e calculating the combinations which have a couple or all singles, the complexity jumps up.
1 - x method
The number of ways = 10C4 - (no of ways to select 2 couple) = 10C4 - 3C2 = 207
Forward method
---------------
No of ways to select one couple and two singles + No of ways to select all singles |
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Re: A group of 10 people [#permalink]
17 May 2011, 09:27
Complicated method -
You have to consider all the possibilities.
Obvious cases -
All 4 singles &
1 married couple and 2 singles
Other cases -
1 marrried person from a couple and 3 singles
1 married couple, 1 married person from a couple and 1 single
1 married couple, 2 married person from different couples
1 married person from each couple and 1 single
1 married person from 2 separate couples and 2 singles
i still dont know if i missing any other case...
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Re: A group of 10 people [#permalink]
17 May 2011, 11:23
well the approach has been discussed over here. permutation-and-combination-help-110982.htmlhope this will help.
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Re: A group of 10 people [#permalink]
17 May 2011, 18:59
fluke, Ian, Karishma anyone? This is how I tried counting it forward. 1 couple and 3 singles --------------------- One couple, 2 singles among rest of couples, 1 singles from 4 3C1 * (6 * 4/2) * 4C1 = 3 * 12 * 4 = 144 One couple, 1 single among rest of couples, 2 singles from 4 3C1 * 6 * 4C2 = 3 * 6 * 6 = 108 One couple, 3 singles from 4 3C1 * 4C3 = 3 * 4 = 12 All 4 singles ------------- 3 singles among 3 couples, 1 single from 4 (6 * 4 * 2) / 3! * 4C1 = 8 * 4 = 32 2 singles among 3 couples, 2 single from 4 (6 * 4)/2 * 4C2 = 12 * 6 = 72 1 single among 3 couples, 3 single from 4 6 * 4C3 = 24 4 singles from 4 4C4 = 1  Adding all the above = 264+133 = 397. Way tooo much than the answer. Can you point the error?
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Re: A group of 10 people [#permalink]
17 May 2011, 21:06
I may be drunk that's why. Thanks so much ! yes it all adds up to sanity i.e. 207 VeritasPrepKarishma wrote: gmat1220 wrote: fluke, Ian, Karishma anyone? This is how I tried counting it forward. 1 couple and 3 singles - C'mon! 1 couple (2 people) + 3 singles? That's 5 people!--------------------- One couple, 2 singles among rest of couples, 1 singles from 4 3C1 * (6 * 4/2) * 4C1 = 3 * 12 * 4 = 144 One couple, 1 single among rest of couples, 2 singles from 4 3C1 * 6 * 4C2 = 3 * 6 * 6 = 108 One couple, 3 singles from 4 3C1 * 4C3 = 3 * 4 = 12 All 4 singles - Looks Fine------------- 3 singles among 3 couples, 1 single from 4 (6 * 4 * 2) / 3! * 4C1 = 8 * 4 = 32 2 singles among 3 couples, 2 single from 4 (6 * 4)/2 * 4C2 = 12 * 6 = 72 1 single among 3 couples, 3 single from 4 6 * 4C3 = 24 4 singles from 4 4C4 = 1 Adding all the above = 264+133 = 397. Way tooo much than the answer. Can you point the error? |
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Re: A group of 10 people [#permalink]
17 May 2011, 21:50
gmat1220 wrote: A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be from the 10 people. How many different committees can be formed if the committee can consist of at most 1 married couple? Can someone employ the forward method i.e. (not the 1-x method to deduce the answer) ? I think I over counting as my answer overshoots the OA Answer = No of ways to select one couple and two singles + No of ways to select all singles OA is 207.
Using 1 - x method the question is piece of cake. But if you employ the forward method i.e calculating the combinations which have a couple or all singles, the complexity jumps up.
1 - x method
The number of ways = 10C4 - (no of ways to select 2 couple) = 10C4 - 3C2 = 207
Forward method
---------------
No of ways to select one couple and two singles + No of ways to select all singles Let A----B C----D E ----F are couples and 1 2 3 4 are singles 1st scenario .. 4 singles out of 4 singles ------- 4c4 = 1 2nd scenario 2 Married singles + 2 singles ------ ( 6*4)/2 * 4c2 = 72 3rd 1 Married single + 3 singles ------- 6*4c3 = 24 4th 3 married single + 1 single ------- (6*4*2)/3!*4C1 = 32 5th scenario 1Couple + 1 Married single + 1 single ---- 3c1*4*4c1= 48 6th scenario 1 couple + 2 Married singles ------- 3c1*( 4*2)/2 = 12 7th scenario 1 couple + 2 singles --------- 3c1*4C2 = 18 Total = 1+72+24+ 32+48+12+18 = 207
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Re: A group of 10 people [#permalink]
14 Sep 2011, 13:24
sudhir18n wrote: gmat1220 wrote: A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be from the 10 people. How many different committees can be formed if the committee can consist of at most 1 married couple? Can someone employ the forward method i.e. (not the 1-x method to deduce the answer) ? I think I over counting as my answer overshoots the OA Answer = No of ways to select one couple and two singles + No of ways to select all singles OA is 207.
Using 1 - x method the question is piece of cake. But if you employ the forward method i.e calculating the combinations which have a couple or all singles, the complexity jumps up.
1 - x method
The number of ways = 10C4 - (no of ways to select 2 couple) = 10C4 - 3C2 = 207
Forward method
---------------
No of ways to select one couple and two singles + No of ways to select all singles Let A----B C----D E ----F are couples and 1 2 3 4 are singles 1st scenario .. 4 singles out of 4 singles ------- 4c4 = 1 2nd scenario 2 Married singles + 2 singles ------ ( 6*4)/2 * 4c2 = 72 3rd 1 Married single + 3 singles ------- 6*4c3 = 24 4th 3 married single + 1 single ------- (6*4*2)/3!*4C1 = 32 5th scenario 1Couple + 1 Married single + 1 single ---- 3c1*4*4c1= 48 6th scenario 1 couple + 2 Married singles ------- 3c1*( 4*2)/2 = 12 7th scenario 1 couple + 2 singles --------- 3c1*4C2 = 18 Total = 1+72+24+ 32+48+12+18 = 207 hi - you could have simply done this .. Total number of ways of selecting 4 people out of 10 = 10C4 Total number of ways of selecting 2 married couple = 3C2 Answer = 10C4 - 3C2 = 210-3 = 207..
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Re: A group of 10 people [#permalink]
10 Oct 2011, 06:32
total number of combinations 10c4 = 10!/4!6! = 210 total number of married couple = 3c2 = 3!/2! = 3 total number of possible commities are 207
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Re: A group of 10 people [#permalink]
10 Oct 2011, 07:39
if we change the question to "...at least 1 married couple", whats gonna be the answer?
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Re: A group of 10 people [#permalink]
11 Oct 2011, 23:36
If its atleast one couple.. then shud be 3C1*4C2 + 3C2 ways = 21 ways
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Re: A group of 10 people [#permalink]
16 Nov 2011, 16:38
We can use the other way of thinking, that is (total possibility of selecting 4 people)-(selecting two couples)
then the answer will be 10C4-3C2=210-3=207. That's much more quickly
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Re: A group of 10 people [#permalink]
13 Feb 2013, 18:27
One way to do problem: Answer = No of ways to select one couple and two singles + No of ways to select all singles=207=official answer How about the 1-x method "The number of ways = 10C4 - (no of ways to select 2 couple) = 10C4 - 3C2 = 207" Since when has the opposite of "at most 1" equaled to exact no of ways to select 2 couples? Could someone help me grasp why this is conceptually the opposite of exactly 1 couple?
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Re: A group of 10 people [#permalink]
13 Feb 2013, 20:31
manimgoindowndown wrote: One way to do problem: Answer = No of ways to select one couple and two singles + No of ways to select all singles=207=official answer
How about the 1-x method
"The number of ways = 10C4 - (no of ways to select 2 couple) = 10C4 - 3C2 = 207"
Since when has the opposite of "at most 1" equaled to exact no of ways to select 2 couples? Could someone help me grasp why this is conceptually the opposite of exactly 1 couple? You need to select a committee of 4. You can do it in 3 ways: a. Select no couple and 4 others b. Select one couple and 2 others c. Select two couples 'At most one' means either no couples or only one couple. What is the complement of at most one? The only other way in which you can select a committee of 4 is by selecting 2 couples. So you can find the number of ways of selecting 2 couples and subtract that from the total number of ways of selecting 4 people. You will get the number of ways in which there will be at most one couple.
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Re: A group of 10 people [#permalink]
15 Feb 2013, 04:47
VeritasPrepKarishma wrote: manimgoindowndown wrote: One way to do problem: Answer = No of ways to select one couple and two singles + No of ways to select all singles=207=official answer
How about the 1-x method
"The number of ways = 10C4 - (no of ways to select 2 couple) = 10C4 - 3C2 = 207"
Since when has the opposite of "at most 1" equaled to exact no of ways to select 2 couples? Could someone help me grasp why this is conceptually the opposite of exactly 1 couple? You need to select a committee of 4. You can do it in 3 ways: a. Select no couple and 4 others b. Select one couple and 2 others c. Select two couples 'At most one' means either no couples or only one couple. What is the complement of at most one? The only other way in which you can select a committee of 4 is by selecting 2 couples. So you can find the number of ways of selecting 2 couples and subtract that from the total number of ways of selecting 4 people. You will get the number of ways in which there will be at most one couple. Namaste aur shukria so much again Karishma. Made it clear within the context of the problem
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Re: A group of 10 people consists of 3 married couples and 4 [#permalink]
16 Feb 2013, 04:10
gmat1220 wrote: A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be from the 10 people. How many different committees can be formed if the committee can consist of at most 1 married couple? Can someone employ the forward method i.e. (not the 1-x method to deduce the answer) ? I think I over counting as my answer overshoots the OA Answer = No of ways to select one couple and two singles + No of ways to select all singles OA is 207.
Using 1 - x method the question is piece of cake. But if you employ the forward method i.e calculating the combinations which have a couple or all singles, the complexity jumps up.
1 - x method
The number of ways = 10C4 - (no of ways to select 2 couple) = 10C4 - 3C2 = 207
Forward method
---------------
No of ways to select one couple and two singles + No of ways to select all singles Similar questions to practice: if-a-committee-of-3-people-is-to-be-selected-from-among-98533.htmlps-combinations-94068.htmlps-combinations-101784.htmlcommittee-of-88772.htmlif-4-people-are-selected-from-a-group-of-6-married-couples-99055.htmlif-there-are-four-distinct-pairs-of-brothers-and-sisters-99992.htmlHope it helps.
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Re: A group of 10 people consists of 3 married couples and 4 [#permalink]
22 Feb 2013, 07:47
my question is : i understand we can do like selecting 2 couples and subtract that from the total number of ways of selecting 4 people. now there 6 married peple and 4 singles so it should be 10c4 - 6x5x4x3/4x3x2x1 but i dont understand how 10c4- 3x2/2 comes?? can someone please explain.
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Re: A group of 10 people consists of 3 married couples and 4 [#permalink]
22 Feb 2013, 08:04
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Re: A group of 10 people consists of 3 married couples and 4 [#permalink]
22 Feb 2013, 08:10
bunnel thanks for kind response. One thing i dont understand is while taking total number of ways of selecting 4 people we count the 3 married couples as 6 right.?so when we are selecting 2 married couples it should be 6x5x4x3/4! na??but why when selecting 2 married couples we are taking count just as 3x2/2!?? please do explain!!!
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Re: A group of 10 people consists of 3 married couples and 4 [#permalink]
22 Feb 2013, 08:13
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Re: A group of 10 people consists of 3 married couples and 4
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22 Feb 2013, 08:13
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