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# combi ques (m08q04)

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01 Aug 2008, 07:01
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4 women and 6 men work in the accounting department. In how many ways can a committee of 3 be formed if it has to include at least one woman?

(A) 36
(B) 60
(C) 72
(D) 80
(E) 100

[Reveal] Spoiler: OA
E

Source: GMAT Club Tests - hardest GMAT questions

select 1 women 4C1 and then selecting 2 from rest of 9 is 9C2 , so total= 4C1 + 9C2

But the answer is 100 .
What Am I doing wrong ?
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01 Aug 2008, 07:05
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The better approach might be to figure out how many different committees can be formed regardless of the rule that it must contain 1 woman.

That could be $$C_{10}^3 = 120$$. Then subtract out the number of committees that do not conform to the rule that at least 1 woman must be on the committee. The committees that do not conform to the rule are the committees that are made up entirely of men. The is represented by $$C_6^3=20$$ because we're figuring out how many ways can you select all 3 from the 6 men present. This gives you 120 - 20 = 100, the answer.

abhaypratapsingh wrote:
4 women and 6 men work in the accounting department. In how many ways can a committee of 3 be formed if it has to include at least one woman?

* 36
* 60
* 72
* 80
* 100
select 1 women 4C1 and then selecting 2 from rest of 9 is 9C2 , so total= 4C1 + 9C2

But the answer is 100 .
What Am I doing wrong ?

_________________

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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$. GMAT Club Premium Membership - big benefits and savings Director Joined: 27 May 2008 Posts: 549 Followers: 8 Kudos [?]: 282 [1] , given: 0 Re: combi ques [#permalink] ### Show Tags 01 Aug 2008, 07:29 1 This post received KUDOS abhaypratapsingh wrote: 4 women and 6 men work in the accounting department. In how many ways can a committee of 3 be formed if it has to include at least one woman? * 36 * 60 * 72 * 80 * 100 select 1 women 4C1 and then selecting 2 from rest of 9 is 9C2 , so total= 4C1 + 9C2 But the answer is 100 . What Am I doing wrong ? the question asks "at least one Woman" so you can have 1W + 2M = 4C1 * 6C2 = 4 * 15 = 60 2W + 1M = 4C2 * 6C1 = 6 * 6 = 36 3W = 4C3 = 4 Total = 60+36+4 = 100 Senior Manager Joined: 06 Apr 2008 Posts: 449 Followers: 1 Kudos [?]: 125 [0], given: 1 Re: combi ques [#permalink] ### Show Tags 01 Aug 2008, 09:32 abhaypratapsingh wrote: 4 women and 6 men work in the accounting department. In how many ways can a committee of 3 be formed if it has to include at least one woman? * 36 * 60 * 72 * 80 * 100 select 1 women 4C1 and then selecting 2 from rest of 9 is 9C2 , so total= 4C1 + 9C2 But the answer is 100 . What Am I doing wrong ? It says atleast 1 woman Total no. of ways = 3 women OR 2women + 1man OR 1woman + 2 men = 4C3 + 4C2*6C1 + 4C1*6C2 = 4 + 36 + 60 = 100 Manager Joined: 21 Mar 2006 Posts: 132 Followers: 3 Kudos [?]: 15 [0], given: 0 Re: combi ques [#permalink] ### Show Tags 01 Aug 2008, 09:46 i understand "Total no. of ways = 3 women OR 2women + 1man OR 1woman + 2 men" but I donot understand why following is wrong : Total = (no. of ways to select one woman ) * (select any two from the rest 9) = 4C1 * 9C2 = 154 Senior Manager Joined: 06 Apr 2008 Posts: 449 Followers: 1 Kudos [?]: 125 [1] , given: 1 Re: combi ques [#permalink] ### Show Tags 01 Aug 2008, 10:04 1 This post received KUDOS abhaypratapsingh wrote: i understand "Total no. of ways = 3 women OR 2women + 1man OR 1woman + 2 men" but I donot understand why following is wrong : Total = (no. of ways to select one woman ) * (select any two from the rest 9) = 4C1 * 9C2 = 154 Because in your method you are double counting Suppose there were four women A, B, C, D and six men 1,2,3,4,5,6 Then according to you if A is selected then committee can be A,B,C as one of option If B is selected using 4C1 then B,A,C can be one of option which is same as above. Similarly A, B , 1 and B, A, 1 will be double count Therefore if you use your method you have to subtract total number of double counts Manager Status: I rest, I rust. Joined: 04 Oct 2010 Posts: 122 Schools: ISB - Co 2013 WE 1: IT Professional since 2006 Followers: 17 Kudos [?]: 117 [3] , given: 9 Re: combi ques [#permalink] ### Show Tags 27 Oct 2010, 06:17 3 This post received KUDOS Ways to select 3 out of 10 individuals (4W+6M) = 10C3 = 120 Ways to select 3 out of 6 men = 6C3 = 20. These 20 are the combinations that include no women. Every other combination must include atleast 1 woman. Therefore, required ways = 120 - 20 = 100. E _________________ Respect, Vaibhav PS: Correct me if I am wrong. Intern Joined: 13 Oct 2010 Posts: 25 Followers: 1 Kudos [?]: 8 [4] , given: 3 Re: combi ques (m08q04) [#permalink] ### Show Tags 27 Oct 2010, 14:48 4 This post received KUDOS The answer is E. Permutations and Combinations is very confusing so I solved this question in this way: Atleast one women gives us 3 scenarios, 1) All 3 women 2) 1 women and 2 men 3) 2 women and 1 man. So for all 3 women we get 4 possible ways. For 1 women and 2 men we get 60 possible ways. FOr 2 women and 1 man we get 36 possible ways. So the total is 4+60+36 = 100. Manager Status: Fighting the beast. Joined: 25 Oct 2010 Posts: 182 Schools: Pitt, Oregon, LBS... Followers: 32 Kudos [?]: 390 [1] , given: 36 Re: combi ques (m08q04) [#permalink] ### Show Tags 28 Oct 2010, 10:38 1 This post received KUDOS Ok guys, i have a question, and forgive me for it being an amateur one. How do you conclude that out of 10 people, there are 120 combinations of 3? _________________ [highlight]Monster collection of Verbal questions (RC, CR, and SC)[/highlight] http://gmatclub.com/forum/massive-collection-of-verbal-questions-sc-rc-and-cr-106195.html#p832142 [highlight]Massive collection of thousands of Data Sufficiency and Problem Solving questions and answers:[/highlight] http://gmatclub.com/forum/1001-ds-questions-file-106193.html#p832133 SVP Joined: 30 Apr 2008 Posts: 1888 Location: Oklahoma City Schools: Hard Knocks Followers: 39 Kudos [?]: 538 [0], given: 32 Re: combi ques (m08q04) [#permalink] ### Show Tags 28 Oct 2010, 10:55 MisterEko wrote: Ok guys, i have a question, and forgive me for it being an amateur one. How do you conclude that out of 10 people, there are 120 combinations of 3? Don't feel bad about asking that question. There no stupid questions, just stupidly phrased questions. Your question is neither. The formula is $$\frac{10!}{3!(10-3)!}$$ This means on top 10*9*8*7*6*5*4*3*2*1 divided by 3*2*1*7*6*5*4*3*2*1. You can do some cancelling out. The top 7 through 1 cancels out with the bottom 7 through 1 leaving $$\frac{10*9*8}{3*2}$$ 10 / 2 = 5, and 9 /3 = 3, so now we don't have a denominator, and we are left with 5*3*8 = 15 * 8 = 120. _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

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31 Oct 2011, 07:02
SPOT THE PATTERN !!

Thanks for this question.

To add usefully to the thread, I would highlight that you should spot the words "at least one" in the question stem.
That should lead you to directly choose the following method: "All possible outcomes - those you don't want", i.e. "all committees - committees with only males".

By doing so, you don't wander trying different things and save some time and calculations.
PS: it just took me 7 minutes working the question before realizing that it can be done in a minute...
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04 Nov 2011, 20:00
You can work this by
Total Ways Of Selecting Commitee - ways of selecting NO women

Using Slot Method
10x9x8/3! - 6x5x4/3!

= 120 - 20

= 100

Hope that helps.
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03 Dec 2011, 11:41
Fairly simple.

First calculate the total of ways to choose 3 out of 10.

So this is 10!/3!(10-3)! which works out to 120.
Then you need to subtract the chances of getting no women - 6!/3!(6-3)! = 20.

Then 120-20 = 100.

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05 Oct 2013, 08:15
abhaypratapsingh wrote:
4 women and 6 men work in the accounting department. In how many ways can a committee of 3 be formed if it has to include at least one woman?

(A) 36
(B) 60
(C) 72
(D) 80
(E) 100

[Reveal] Spoiler: OA
E

Source: GMAT Club Tests - hardest GMAT questions

select 1 women 4C1 and then selecting 2 from rest of 9 is 9C2 , so total= 4C1 + 9C2

But the answer is 100 .
What Am I doing wrong ?

I followed the exact method.. I still can't understand what I did wrong..???

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17 Oct 2013, 04:12
domfrancondumas wrote:
abhaypratapsingh wrote:
4 women and 6 men work in the accounting department. In how many ways can a committee of 3 be formed if it has to include at least one woman?

(A) 36
(B) 60
(C) 72
(D) 80
(E) 100

[Reveal] Spoiler: OA
E

Source: GMAT Club Tests - hardest GMAT questions

select 1 women 4C1 and then selecting 2 from rest of 9 is 9C2 , so total= 4C1 + 9C2

But the answer is 100 .
What Am I doing wrong ?

I followed the exact method.. I still can't understand what I did wrong..???

The number you get will have duplications.

Consider the group of three women {ABC}
Say you select women A with 4C1, next you can get women B and C with 9C2: {A}{BC}. Now, with the same method you can get the following group {B}{AC}, which is basically the same group.

Does this make sense?
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29 Oct 2013, 07:55
What is wrong in this method:

Case I : 1 woman + 2 Men , we have three places to fill _ _ _

I fill first place with a woman, so first place can be filled with 4 ways because there are 4 women, 2nd place can be filled with 6 ways because there are 6 men, third place can be filled with 5 men so in case I total ways : 4 x 6 x 5 = 120

Case II : 2 women + 1 men, three places to fill _ _ _

Lets place 2 women first. so first place can be filled with 4 ways, 2nd place can be filled with 3 ways and 3rd place can be filled with 6 ways (6 men) so

4 X 3 X 6 = 72

Case III : 3 women , three places _ _ _

So 4 X 3 X 2 = 24

So total ways should be : 120 + 72 + 24 = 216

Can anybody please explain what is wrong in this method ?
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29 Oct 2013, 08:54
dheeraj787 wrote:
What is wrong in this method:

Case I : 1 woman + 2 Men , we have three places to fill _ _ _

I fill first place with a woman, so first place can be filled with 4 ways because there are 4 women, 2nd place can be filled with 6 ways because there are 6 men, third place can be filled with 5 men so in case I total ways : 4 x 6 x 5 = 120

Case II : 2 women + 1 men, three places to fill _ _ _

Lets place 2 women first. so first place can be filled with 4 ways, 2nd place can be filled with 3 ways and 3rd place can be filled with 6 ways (6 men) so

4 X 3 X 6 = 72

Case III : 3 women , three places _ _ _

So 4 X 3 X 2 = 24

So total ways should be : 120 + 72 + 24 = 216

Can anybody please explain what is wrong in this method ?

dude you need not use permutations here, you just have you make combinations. 1. 1 woman and 2 men - 4c1x6c2= 60 2. 2 women and 1 men - 4c2x6c1= 36. 3. all three women- 4c3= 4. Add em all to get 100 cases
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29 Oct 2013, 10:12
lepatron wrote:
SPOT THE PATTERN !!

Thanks for this question.

To add usefully to the thread, I would highlight that you should spot the words "at least one" in the question stem.
That should lead you to directly choose the following method: "All possible outcomes - those you don't want", i.e. "all committees - committees with only males".

By doing so, you don't wander trying different things and save some time and calculations.
PS: it just took me 7 minutes working the question before realizing that it can be done in a minute...

Great tip, recognizing that you should subtract combinations with all men instead of adding combinations with women is key.
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29 Oct 2013, 23:56
At least 1 means '1 or more'.

In this case, there are 4 women to choose from.

The possible methods of forming 'committees of 3' with AT LEAST 1 WOMAN are:

(1) 3 Women & 0 Men ———> 4C3 * 6C0 = 4 * 1 = 4

OR

(2) 2 Women & 1 Man ———> 4C2 * 6C1 = 6 * 6 = 36

OR

(3) 1 Woman & 2 Men ———> 4C1 * 6C2 = 4 * 15 = 60

So, there are 3 possible methods of forming committees. The 3 answers have to be added to arrive at the final answer.

Adding all the possible options --> 4 + 36 + 60 = 100

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02 Nov 2013, 13:42
thanks helped a lot
Re: combi ques (m08q04)   [#permalink] 02 Nov 2013, 13:42
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