Team A and Team B are competing against each other in a game

Question

Team A and Team B are competing against each other in a game of tug-of-war. Team A, consisting of 3 males and 3 females, decides to lineup male, female, male, female, male, female. The lineup that Team A chooses will be one of how many different possible lineups?

(A) 9
(B) 12
(C) 15
(D) 36
(E) 720

(A) 9
(B) 12
(C) 15
(D) 36
(E) 720

Elite097 · Answer

GMATinsight   avigutman   ScottTargetTestPrep  The question is worded as "chooses will be one of how many different possible lineups" so clearly mfmfmf can be one of 6!/3!3! lineups. I agree they have chosen mfmfmf but there is no reason to think that it means 1 specific permutation in the order mfmfmf. Ex. M1F1M2F2M3F3 being one specific permutation of mfmfmf, which can be calculated in 3!3! ways.

It can easily mean that the arrangement mfmfmf is one out of how many total arrangements i.e 6!/3!3!? Pls clarify as no explanation here seems to do the job. It easily seems to mean that mfmfmf is the chosen lineup among other possibilities

avigutman · Answer

Hi  Elite097 . It seems that you're struggling not with the quantitative reasoning here, but rather with the reading comprehension. 
 
In the question stem quote above, I bold-faced three words in the hopes that emphasizing them will assist you.
Once the team *has made the decision* to lineup in the particular configuration of mfmfmf, the number of possible lineups *in that particular configuration* is 3!3!.
Please let me know if you have any follow-up questions.

ThatDudeKnows · Answer

Elite097

What if I told you that out of the 262 Problem Solving questions in the 2022 OG, the 212 Problem Solving questions in the 2022 Quant Review, the 240 Data Sufficiency questions in the 2022 OG, and the 161 Data Sufficiency questions in the 2022 Quant Review, I'm pretty sure that exactly ZERO require you to actually know or use nCr OR nPr. That's 875 official questions on which they could have asked a question of that kind. But they never did. Not once.  If you find one, please let me know!!

Until GMAC goes back to including even a single official question that requires either of those, my advice is not to bother with them. There are a bunch of ways those two things can be used and the number of twists and turns in a question can definitely make it so you need to understand far more nuances than are worthwhile. Every Perm/Comb question in the 2022 editions can be solved withOUT knowing anything about nCr or nPr. Formulas are great IF you know exactly when and how they can be used (and when and why they can't!), but why bother?

For this question (and for the vast majority of Perm/Comb questions), I think you're better off just reasoning through it rather than trying to employ some formula that you've memorized without TRULY understanding.  BrentGMATPrepNow  hit the nail on the head on this problem (and the videos in his post are definitely worth a watch if you're missing questions like this). Just work through the line in order.

First person has to be a man. How many options do we have? 3
Second person has to be a woman. How many options do we have? 3
Third has to be a man; we have two men remaining. How many options do we have? 2
Fourth has to be a woman; we have two women remaining. How many options do we have? 2
Fifth has to be a man; we have one man remaining. How many options do we have? 1
Sixth has to be a woman; we have one woman remaining. How many options do we have? 1

3*3*2*2*1*1 = 36

Answer choice D.

Sanaya24 · Answer

This question is really confusing, it states that Team A chooses a line up of M-F-M-F-M-F and then says this line up is one of how many different possible lineups.. Doesn't that mean that this is technically 1 of 6! line ups? I'm confused, any help will be appreciated!!

Bunuel · Answer

It seems you did not read the question and provided solutions carefully. The lineup is fixed as M-F-M-F-M-F. However, the men and women themselves, in their respective places, can be arranged in different ways: 3! ways for the men and 3! for the women. This results in a total of 3!*3! = 36 specific lineups.

DanTheGMATMan · Answer

Draw slots and fill in your options like any permutation:

https://www.youtube.com/watch?v=YclVe6bLDbI

jayakalra · Answer

For anyone who has the doubt, why 720 cannot be the answer, I have spent quiet some time decoding that too. So according to my understanding, if you're someone who is selecting 6! or 720 as the answer you might be interpreting that the question is asking the total number of arrangements of lining up 3 males and 3 females. But imagine this
720 would include cases where mmfmff or mmmfff or mfmffm etc. are also possible, which is 6!/3!*3! = 20 , so for every 20 line ups there will be 36 ways (3!*3!) ways in which the males and females will be arranged.

So either it was interpreted as mfmfmf out of 20 different combinations or 36 ways of arranging mfmfmf.
20 is anyway to part of the answer choices.
Hence Ans D.

jafeezy · Answer

This question is very confusing and its because its hard to determine what the question means by the term "lineups" at the end of the last sentence. Is "lineups" referring to all the different ways that the specific MALES and FEMALES can be arranged in the predetermined MFMFMF set up, or does "lineups" mean all the possible ways to arrange the genders, as in MMMFFF, MFMMFF, etc. It doesn't explicitly say which "lineups" its referring to.

"The lineup that Team A chooses will be one of how many different possible lineups" could just as easily mean TEAM A chose MFMFMF out of all the possible lineups that the genders could have been arranged. They chose MFMFMF out of MMFFMF, MMMFFF, FFFMMM, etc. and how many of those COULD they have chosen?

OR

It could mean that TEAM A already chose MFMFMF as their predetermined set up and the question wants to know how many different arrangements you can put the specific males and females in that specific "lineup".

IMO the language isn't clear.

saransh2797 · Answer

Hey Bunuel, A small clarification is required please - In GMAT, how do we infer whether the objects provided in the question stem are identical or not. For ex- in here, it could have been the case that all 3 Males are identical & 3 Females are identical. Does this mean, the question will explicitly state that Males are identical and so on? I face this constraint confusion always whenever objects are involved which may or may not be identical. Since we only divide when there is an identical number of objects. If 3 males and 3 females were identical, we would have divided by 3!3!. Please guide me here. Thanks a ton

Bunuel · Answer

Where clarification is necessary, it will be given. If it's not explicitly stated and the context doesn’t make it clear, use common sense: people are distinct, so their individual positioning changes the lineup, meaning order matters.

SamAn12213 · Answer

The question is relatively easy, could be solved in 10 seconds , use fill in the blanks methods,,,, MFMFMF,,,,, so 3 gaps of M, 3 of F,,,,3*3*2*2*1*1=36 ways simple

PixelVoyager99 · Answer

To determine the number of possible lineups for Team A, where 3 males (M) and 3 females (F) are arranged in an alternating pattern (M, F, M, F, M, F), we can follow these steps:
 
  Arrange the Males:  The 3 males can be arranged in the 3 male positions in 3!3!3! (3 factorial) ways. Calculating 3!3!3!:
3!=3×2×1=63! = 3 	imes 2 	imes 1 = 63!=3×2×1=6 
  Arrange the Females:  Similarly, the 3 females can be arranged in the 3 female positions in 3!3!3! ways. Calculating 3!3!3!:
3!=3×2×1=63! = 3 	imes 2 	imes 1 = 63!=3×2×1=6 
  Total Arrangements:  Since the arrangements of males and females are independent, the total number of lineups is the product of the two arrangements:
3!×3!=6×6=363! 	imes 3! = 6 	imes 6 = 363!×3!=6×6=36  
Therefore, the total number of different possible lineups for Team A is  36 .
 Answer:  (D) 36

tryorcry · Answer

This Question is very confusing. One could find a difference between 
"how many different possible lineups?" and "how many different possible lineups from the given lineup?"

Can one tell if this is an actual gmat question please?

Bunuel · Answer

Yes, this is an official GMAT question: https://gmatclub.com/forum/download/file.php?id=88233 Please review the discussion. This question has already been discussed in quite a bit of detail in the thread. GMAT-Club-Forum-6h0ceet9.png

Adit_ · Answer

I understand this is an official question but I still have my doubts.
It says one of how many possible lineups and calls the aforementioned lineup as "one". 
So one of how many total could mean how many such blueprints possible? Or how many kinds of lineups possible with one being M,F,M,F,M,F?
In that case shouldn't it be 6!/3!*3! = 20? Kindly help me out.

PS:- Yes I read the thread as well but I am not satisfied.
Let me try rephrasing it again:
" The lineup that Team A chooses will be one of how many different possible lineups?"
This in other words IMO says, the lineup that team A chooses = M,F,M,F,M,F is one out of how many different "possible" lineups.
It calls that setup a "lineup" so the number of lineups must technically be the numbers of ways of having an M,F combo arranged in different ways right?
If it were like how the answer is, it should be like how many different possible arrangement of the lineup are there, or something like that?

ShreyaMittal1609 · Answer

I was also confused by the wording. But I was sure that it won't be 6! ways because MMMFFF can't be arranged in 6! ways considering we only want to determine the lineup (sequence of Males and females).

I thought it would be 6!/(3!*3!) = 20

But this option is not given.

After reading the discussion, my understanding is :- 
The question stem- The lineup that Team A chooses will be one of how many different *possible lineups*? Question is asking different ways of arranging people in the chosen lineup (MFMFMF).

pusansen99 · Answer

For anyone still struggling with the wording or trying to visualise this question, solving it  slot-by-slot  using the  Fundamental Counting Principle (FCP)  makes the constraint crystal clear.
The prompt states the team decides to line up in this exact pattern:  M - F - M - F - M - F . This fixes the gender roles for all 6 spots.
Let's assume our 3 unique males are ( M_1, M_2, M_3 ) and 3 unique females are ( F_1, F_2, F_3 ). Now, let's fill the slots one by one from left to right:
 
  Slot 1 (Must be Male):  We have  3  choices available ( M_1, M_2,  or  M_3 ). 
  Slot 2 (Must be Female):  We have  3  choices available ( F_1, F_2,  or  F_3 ). 
  Slot 3 (Must be Male):  Since 1 man is already placed in Slot 1, we have  2  choices left. 
  Slot 4 (Must be Female):  Since 1 woman is already placed in Slot 2, we have  2  choices left. 
  Slot 5 (Must be Male):  Two men are already in line, leaving us with  1  choice. 
  Slot 6 (Must be Female):  Two women are already in line, leaving us with  1  choice.  
Multiplying the independent choices for each slot together:
 3 * 3 * 2 * 2 * 1 * 1 = 36 
 Why this works: 
If you group the numbers by gender, it rearranges beautifully into the standard factorial approach:
 (3 * 2 * 1) * (3 * 2 * 1) = 3! * 3! = 36 
The slot method explicitly shows how the pool of unique individuals shrinks as you move down the line, making it impossible to count non-alternating patterns like MMMFFF accidentally.
Hope this helps clear up the ambiguity!

minionbaba · Answer

To all those who are confused, this will help"

An analogy

Suppose I say:

"Six books are to be arranged on a shelf, and I decide that the first three positions will contain only science books and the last three only history books. How many different arrangements are possible?"

You would not count all 6!=720 permutations, because this violates my earlier stated decision. You would count: 3!×3!=36.

PGTLrowanhand · Answer

Here's my video solution:  https://www.youtube.com/watch?v=wSj-oHJmI8s

Carcass · Answer

Long time I did not see you  PGTLrowanhand . Great GMAT tutor

Team A and Team B are competing against each other in a game

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Team A and Team B are competing against each other in a game

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards