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• ### $450 Tuition Credit & Official CAT Packs FREE November 15, 2018 November 15, 2018 10:00 PM MST 11:00 PM MST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) # A firm has at least 5 partners, of whom at least 2 are male and at  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags GMAT Tutor Joined: 15 Nov 2013 Posts: 65 A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 12 Nov 2017, 08:27 3 7 00:00 Difficulty: 95% (hard) Question Stats: 37% (02:02) correct 63% (01:43) wrong based on 224 sessions ### HideShow timer Statistics Time for a brand-new quant problem, ladies and gentlemen! __ A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm’s partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected? (1) The firm has no more than 3 male partners. (2) The firm has no more than 3 female partners. __ It appears that the "delayed OA" option has disappeared, so, the OA for this problem is available immediately. I'll post a detailed solution in a day or two. _________________ Please visit my corners of the web: http://www.RonPurewal.com http://www.facebook.com/RonPurewalGMAT __ Pueden hacerle preguntas a Ron en castellano Potete fare domande a Ron in italiano On peut poser des questions à Ron en français Voit esittää kysymyksiä Ron:lle myös suomeksi __ Un bon vêtement, c'est un passeport pour le bonheur. —Yves Saint-Laurent Senior Manager Joined: 29 Jun 2017 Posts: 468 GPA: 4 WE: Engineering (Transportation) A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 12 Nov 2017, 10:18 Hello ron I am feeling happy to reply on your post in first comment My solution : 1) consider MM FFF 5 members and MMFFFF 6 members MMFFF probability equal MMFF => 2C2.3C2 / 5C4 = 3/5 MMFFFF probability equal MMFF => 2C2.4C2/ 6C4 = 2/5 A and D are out 2) Same case apply for 2nd point The firm has no more than 3 female partners. just replace M by F and F by M in part 1) so B is also out Combine 1 and 2 Possibilities MMFFF and FFMMM case a MMMFF and FFFMM case b consider both case now case a MMFFF => probability is 3/5 calculated above , holds good for FFMMM case b MMMFF => its same case as later part of case a , which gives 3/5 and same is FFFMM which is first part of case a so combination of 1 and 2 gives => probability 3/5. IMO C _________________ Give Kudos for correct answer and/or if you like the solution. GMAT Tutor Joined: 15 Nov 2013 Posts: 65 Re: A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 12 Nov 2017, 12:00 1 sahilvijay — it seems you've forgotten to consider "MMMFFF" — i.e., 3 men and 3 women — as a pool from which to choose. This, too, is a possibility under statements 1 and 2 together — and so you can't conclude C without verifying that it gives the same probability as the cases you actually have considered. (You've considered only "2 men, 3 women" and "3 men, 2 women" as possible pools for the two statements together.) _________________ Please visit my corners of the web: http://www.RonPurewal.com http://www.facebook.com/RonPurewalGMAT __ Pueden hacerle preguntas a Ron en castellano Potete fare domande a Ron in italiano On peut poser des questions à Ron en français Voit esittää kysymyksiä Ron:lle myös suomeksi __ Un bon vêtement, c'est un passeport pour le bonheur. —Yves Saint-Laurent Senior Manager Joined: 29 Jun 2017 Posts: 468 GPA: 4 WE: Engineering (Transportation) Re: A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 12 Nov 2017, 12:02 Yes , i forgot this case , my bad Sent from my iPhone using GMAT Club Forum _________________ Give Kudos for correct answer and/or if you like the solution. Senior Manager Joined: 29 Jun 2017 Posts: 468 GPA: 4 WE: Engineering (Transportation) Re: A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 12 Nov 2017, 12:32 Adding to my solution stated above, one more case , which i forgot to consider With MMMFFF -> 3C2.3C2 / 6C4 3x3/15 3/5 Therefore C Posted from my mobile device _________________ Give Kudos for correct answer and/or if you like the solution. GMAT Tutor Joined: 15 Nov 2013 Posts: 65 Re: A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 12 Nov 2017, 12:51 1 Yep. You should ALSO work this problem by LISTING OUT THE CASES for the "MMFFF" and "MMMFFF" pools — and verifying that exactly three-fifths of the possibilities in each list comprise 2 men and 2 women. (Having made the list for "MMFFF", you don't need to make the list for "MMMFF"; that would just be the same list, with all the M's and F's reversed. Since the desired outcome is symmetric with regard to M's and F's, the probability will definitely be the same if all the M's and F's are switched.) _________________ Please visit my corners of the web: http://www.RonPurewal.com http://www.facebook.com/RonPurewalGMAT __ Pueden hacerle preguntas a Ron en castellano Potete fare domande a Ron in italiano On peut poser des questions à Ron en français Voit esittää kysymyksiä Ron:lle myös suomeksi __ Un bon vêtement, c'est un passeport pour le bonheur. —Yves Saint-Laurent Math Expert Joined: 02 Aug 2009 Posts: 7025 A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 12 Nov 2017, 18:21 RonPurewal wrote: Time for a brand-new quant problem, ladies and gentlemen! __ A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm’s partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected? (1) The firm has no more than 3 male partners. (2) The firm has no more than 3 female partners. __ It appears that the "delayed OA" option has disappeared, so, the OA for this problem is available immediately. I'll post a detailed solution in a day or two. individually statements are not sufficient... so lets look at it when combined.... M and F can be 2 or 3 so total can be 5 or 6 possibilities- 1) MMFFF 2)MMMFF 3) MMMFFF Now generally ans would be E as different possibilities are there But can be SUFFICIENT if all have same probability MMFFF and FFMMM will be same equal :- 2 M are chosen and 2 are to be picked up from 3 F = $$3C2 = 3$$ total :- 5C4 = 5 $$P = \frac{3}{5}$$ lets check MMMFFF choosing 4 :- 2 out of 3 M and 2 out of 3 F = $$3C2*3C2 = 3*3=9$$ total ways :- 4 out of 6 = $$6C4 = \frac{6!}{4!2!} = \frac{6*5}{2}=15$$ $$P = \frac{9}{15} = \frac{3}{5}$$ so ans is $$\frac{3}{5}$$ in all cases sufficient _________________ 1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html GMAT online Tutor GMAT Tutor Joined: 15 Nov 2013 Posts: 65 Re: A firm has at least 5 partners, of whom at least 2 are male and at [#permalink] ### Show Tags 13 Nov 2017, 00:09 1 Here's a KEY that uses EXHAUSTIVE LISTS. Even if you solved this problem with combinatorial formulas, you should ALSO practice MAKING EXHAUSTIVE LISTS when you REVIEW it! __ Statement (1) Together with the prompt, this statement tells us that the firm has either 2 or 3 male partners. There is no maximum, however, on the number of female partners; we know only that there are at least 2 female partners (and there can’t be just 2 of each, since the firm has at least 5 partners). Let’s test cases: • If there are 3 female partners (call them ‘1’, ‘2’, ‘3’) and 2 male partners (‘A’, ‘B’), then there are five ways to select 4 of them all. The selections consisting of 2 women and 2 men are boldfaced and asterisked: 1, 2, 3, A 1, 2, 3, B 1, 2, A, B* 1, 3, A, B* 2, 3, A, B* Exactly three of these selections (the last three listed) are made up of 2 women and 2 men. So in this case the desired probability is 3/5. • If there are 4 female partners (‘1’, ‘2’, ‘3’, ‘4’) and 2 male partners (‘A’, ‘B’), then there are 15 ways to select four of the six. The selections consisting of 2 women and 2 men are boldfaced and asterisked: 1, 2, 3, 4 1, 2, 3, A 1, 2, 3, B 1, 2, 4, A 1, 2, 4, B 1, 2, A, B* 1, 3, 4, A 1, 3, 4, B 1, 3, A, B* 1, 4, A, B* 2, 3, 4, A 2, 3, 4, B 2, 3, A, B* 2, 4, A, B* 3, 4, A, B* Of the 15 possible selections, exactly 6 contain equal numbers of women and men (two of each). The desired probability in this case is thus 6/15 = 2/5. These two cases produce two different answers to the problem, so, Statement (1) is NOT SUFFICIENT. __ Statement (2) This time, we know that there are either 2 or 3 female partners, while the number of male partners can be any integer from 2 onward. These are precisely the same numerical restrictions as in Statement (1), except with men and women reversed. This statement is thus functionally identical to Statement (1) and is therefore NOT SUFFICIENT as well. (Just switch men and women in the workup above. With 3 male and 2 female partners, the probability is 3/5; with 4 male and 2 female partners, the probability is 2/5.) __ Statements (1) and (2) TOGETHER With both statements in force, there are only three possibilities for the partners overall: • 3 women and 2 men • 2 women and 3 men • 3 women and 3 men In either of the first two cases — as demonstrated under Statement (1) — the desired probability is 3/5. The only case left to consider is that involving 3 women (‘1’, ‘2’, ‘3’) and 3 men (‘A’, ‘B’, ‘C’). There are, again, 15 ways to choose four of the six. (Note that you don’t have to do this work again from scratch! You can simply copy the list made above for 4 women and 2 men, substituting ‘C’ for ‘4’.) 1, 2, 3, C 1, 2, 3, A 1, 2, 3, B 1, 2, C, A* 1, 2, C, B* 1, 2, A, B* 1, 3, C, A* 1, 3, C, B* 1, 3, A, B* 1, C, A, B 2, 3, C, A* 2, 3, C, B* 2, 3, A, B* 2, C, A, B 3, C, A, B Of the 15 possible selections, exactly 9 contain equal numbers of women and men (two of each). The desired probability in this case is thus 9/15 = 3/5. The probability is thus 3/5 for all cases possible under both statements. The two statements TOGETHER are thus SUFFICIENT. The correct answer is C. _________________ Please visit my corners of the web: http://www.RonPurewal.com http://www.facebook.com/RonPurewalGMAT __ Pueden hacerle preguntas a Ron en castellano Potete fare domande a Ron in italiano On peut poser des questions à Ron en français Voit esittää kysymyksiä Ron:lle myös suomeksi __ Un bon vêtement, c'est un passeport pour le bonheur. —Yves Saint-Laurent Manager Joined: 20 Feb 2017 Posts: 166 Location: India Concentration: Operations, Strategy WE: Engineering (Other) A firm has at least 5 partners, of whom at least 2 are male [#permalink] ### Show Tags 27 Jul 2018, 22:03 A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm's partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected? 1. The firm has no more than 3 male partners 2. The firm has no more than 3 female partners _________________ If you feel the post helped you then do send me the kudos (damn theya re more valuable than$)

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A firm has at least 5 partners, of whom at least 2 are male  [#permalink]

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27 Jul 2018, 22:14
A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm's partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected?

1. The firm has no more than 3 male partners
2. The firm has no more than 3 female partners

To find the probability of selecting equal number , we need to know the gender of 5th member !!

Given:-
Total of Male + Female is atleast 5, It can be more as well !!
2 Male & 2 Female confirmed !! No idea about the 5th member or other members (if any) ?

Statement(1) :-
The firm has no more than 3 male partners.
This statement only tells us, that Male <= 3 , Now male can be 2 or 3 !!
Statement(1) is insufficient !!

Statement(2) :-
The firm has no more than 3 female partners.
This statement only tells us, that Female <= 3 , Now Female can be 2 or 3 !!
Statement(2) is again insufficient !!

Combining both statements :-
Male can be either 2 or 3.
Female can be either 2 or 3 .

Case (1) :- Male 2 , Female 3
Case (2) :- Male 3 , Female 2
Case(3) :- Male 3 , Female 3

The probabilities of selecting 4(2 Male / 2 Female) or 6 (3 Male/ 3 Female) equal number of male and female partner will differ for the above cases !!

Hence Statement(1) & (2) together are insufficient !!
Correct Option :- (E)
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Re: A firm has at least 5 partners, of whom at least 2 are male and at  [#permalink]

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28 Jul 2018, 01:14
Raksat wrote:
A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm's partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected?

1. The firm has no more than 3 male partners
2. The firm has no more than 3 female partners

Merging topics. Please search before posting a question.
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Re: A firm has at least 5 partners, of whom at least 2 are male and at  [#permalink]

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28 Jul 2018, 10:41
RonPurewal wrote:
Time for a brand-new quant problem, ladies and gentlemen!

__

A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm’s partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected?

(1) The firm has no more than 3 male partners.

(2) The firm has no more than 3 female partners.

__

It appears that the "delayed OA" option has disappeared, so, the OA for this problem is available immediately. I'll post a detailed solution in a day or two.

IMO C

1) possibilities:

3M, 4F
2M,3F

In each case, probability is different

2)possibilities:

3F, 2M
2F, 3M

In each case probability is different

Combining:
There are 3 possibilities:
a) 3M, 2F b) 2M, 3F c) 3M,3F

In each of the case, probability is 3/5. Hence C is the answer.
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Re: A firm has at least 5 partners, of whom at least 2 are male and at  [#permalink]

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29 Jul 2018, 18:19
RonPurewal wrote:
A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm’s partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected?

(1) The firm has no more than 3 male partners.

(2) The firm has no more than 3 female partners.

I received a PM requesting that I comment.

Statement 1:
Case 1: Exac
tly 2 female partners
Case 2: 1,000,000 female partners
Clearly, the probability that exactly 2 women will be selected is higher in Case 2 than in Case 1.
INSUFFICIENT.

Statement 2:
Case 1: Exactly 2 male partners
Case 2: 1,000,000 male partners
Clearly, the probability that exactly 2 men will be selected is higher in Case 2 than in Case 1.
INSUFFICIENT.

Statements combined:
A favorable outcome will be yielded if 2 men and 2 women are selected:
MMWW
The 2 men and 2 women can be selected in ANY ORDER.
For this reason, the last step in each of the 3 cases below is to multiply by the number of orderings for the 4 letters MMWW.
Number of ways to arrange 4 elements = 4!.
But when an arrangement includes IDENTICAL elements, we must divide by the number of ways each set of identical elements can be ARRANGED.
The reason:
When the identical elements swap positions, the arrangement doesn't change.
Here, we must divide by 2! to account for the two identical M's and by another 2! to account for the two identical W's.
Thus:
Number of ways to arrange the 4 letters MMWW = 4!/(2!2!) = 6.

Case 1: The partners are composed of 2 men and 3 women, for a total of 5 partners
P(MMWW) $$= \frac{2}{5} * \frac{1}{4} * \frac{3}{3} * \frac{2}{2} = \frac{1}{10}$$
Multiplying the result above by 6, we get:
$$\frac{1}{10} * 6 = \frac{6}{10} = \frac{3}{5}$$

Case 2: The partners are composed of 3 men and 2 women, for a total of 5 partners
P(MMWW) $$= \frac{3}{5} * \frac{2}{4} * \frac{2}{3} * \frac{1}{2} = \frac{1}{10}$$
Multiplying the result above by 6, we get:
$$\frac{1}{10} * 6 = \frac{6}{10} = \frac{3}{5}$$

Case 3: The partners are composed of 3 men and 3 women, for a total of 6 partners
P(MMWW) $$= \frac{3}{6} * \frac{2}{5} * \frac{3}{4} * \frac{2}{3} = \frac{1}{10}$$
Multiplying the result above by 6, we get:
$$\frac{1}{10} * 6 = \frac{6}{10} = \frac{3}{5}$$

Since the resulting probability in each case is THE SAME -- $$\frac{3}{5}$$-- the two statements combined are SUFFICIENT.

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Re: A firm has at least 5 partners, of whom at least 2 are male and at &nbs [#permalink] 29 Jul 2018, 18:19
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