A few men and a few women are seated in a row. The number of men is on

Question

A few men and a few women are seated in a row. The number of men is one greater than the number of women. What is the total number of people?

(1) The people are seated so that no two women sit beside one another.
(2) The number of ways of arranging the seating positions of the men and women is 3(5!)(7!).

shariq41 · Answer

Answer = C

Explanation :-
Let the number of women be x. Therefore, the number of men is x+1.

1) We cannot find the total number of people just by using Statement 1.
For example :-
 i) There are 3 men and 2 women such that : MWMMW.
ii) There are 4 men and 3 women such that : MWMWMWM

Therefore Statement 1 is not sufficient to answer the question.

2) Just by using Statement 2, there could be multiple scenarios where we could arrange in the x+1 men and x women in 3(5!)(7!).
Hence we cannot obtain unique value of the total number of people (that is sum of number of men and women).

However, using Statement 1 and Statement 2, the number of ways of arranging x women and x+1 men such that no two women are together are:-

(x+2)C(x) . x! . (x+1)! = 3.(5!).(7!)

By solving the above equation, we get x=5, which is the number of women.
Number of men = x+1 = 6.
Hence, total number of people = 11.

Therefore, answer is C.

jokschmer · Answer

in the above explanation can you shed some more light on term (x+2)C(x). how you find this or what it denotes.

manishtank1988 · Answer

Vyshak ,  VeritasPrepKarishma ,  Skywalker18 ,  mikemcgarry ,  Abhishek009 ,  msk0657 ,  abhimahna ,  Bunuel ,  Engr2012 
Can someone explain how this highlighted line be calculated?
Thanks

mikemcgarry · Answer

Dear manishtank1988 , I'm happy to respond. :-)

My friend, here's how I would think through the problem. Statement #1: a seating constraint, but no number specific. Insufficient by itself. Statement #2: a number given, but the number of possible seating constraints (e.g women must sit in clusters of 2 or 3 only ) is mind-boggling. Insufficient by itself. Combined: we have a clear seating constraint, which will determine a specific number of seating arrangements for each number of people, and the former will increase rapidly for each one we add to the latter. Thus, specifying a particular number of seating arrangements determines the number of people. Sufficient together. OA = (C) This is GMAT DS, so on principle, I would refuse to do a single calculation in answering the question. Nevertheless, you would like to see the calculation. Again, here's how I would think about it. Let's say there are x women. Just for the visual, I am showing the x = 5 case. These are spaced apart: _ _ _ F _ _ _ F _ _ _ F _ _ _ F _ _ _ F _ _ _ There must be at least one man in each of the (x - 1) spaces between them. Put (x - 1) men there. _ _ _ F _ M _ F _ M _ F _ M _ F _ M _ F _ _ _ We have two men left. Where do we put these last two men? We have (x - 1) slots between the two women as well as two on the ends, for (x + 1) total slots. We could either put both of these two extra men together, or we could put them in two different slots. If they are together, we could put them in (x + 1). Possible outcomes: M M _ F _ M _ F _ M _ F _ M _ F _ M _ F _ _ _ _ _ _ F _ M _ F _ M M M _ F _ M _ F _ M _ F _ _ _ _ _ _ F _ M _ F _ M _ F _ M _ F _ M _ F _ M M If we put the two extra men in two different slots, we would have (1/2)(x+1)(x) choices. Possible outcomes: _ M _ F _ M _ F _ M _ F _ M M _ F _ M _ F _ _ _ _ _ _ F _ M M _ F _ M _ F _ M _ F _ M M _ F _ _ _ Total number of "men slots" = (x + 1) + (1/2)(x + 1)x Multiply that by (x+ 1)!, and then multiply this by x! for the women. I get a different equation. I get total number of arrangements = *(x+ 1)!*x! If x = 5, then this becomes total number of arrangements = (6 + (1/2)(6)(5))*6!*5! = 21*6!*5! = 3*7*6!*5! = 3*7!*5! = 3(5!)(7!) Does all this make sense? Mike :-)

manishtank1988 · Answer

mikemcgarry
First of all, thanks a lot for responding. I appreciate your help.
But i still have a few questions -
[Q-1] I can understand (x+1) => 1st M of the remaining 2 Ms can take (x+1) positions and 2nd M can take [(x+1)-1] = x positions. I don't understand why are we getting this (1/2)?

If we put the two extra men in two different slots, we would have (1/2)(x+1)(x) choices. Possible outcomes:
_ M _ F _ M _ F _ M _ F _ M M _ F _ M _ F _ _ _
_ _ _ F _ M M _ F _ M _ F _ M _ F _ M M _ F _ _ _

[Q-2] Also, i am assuming that we are multiplying by
(x+1)! => because finally there are (x+1) seats for men and we can arrange them in (x+1)! ways &
x! => because finally there are x seats for women and we can arrange them in x! ways
Am i correct in assuming this?
Thanks and looking forward to your reply.

mikemcgarry · Answer

Dear manishtank1988 , I'm happy to respond. :-)

For Q1, We are going to put the two men in two different slots, and there are (x + 1) slots. It's perfectly true we have (x + 1) choices for the first slot, and x choices for the second slot. Let's say that x = 10, so we have 11 choices for the first slot and 10 choices for the second slot. Think about these two particular results: Result #1: first choice: seventh slot; second choice: second slot Result #2: first choice: second slot; second choice: seventh slot These two different choices result in the same outcome, because order of picking the slots doesn't matter--all that matters is the two resultant slots. In fact, every single result is counted twice, once where the leftmost slot was counted before the rightmost slot, and vice versa. We divide by two to eliminate the duplicates. That's why there has to be a 1/2 in the formula. In this situation, the result would be (1/2)(11)(10) = 11*5 = 55. In fact, think about the combinations formula. nCr = frac{n!}{r!(n-r)!} For the r = 2 case, this becomes nC2 = frac{n!}{2(n-2)!} For n = 11, this becomes nC2 = frac{11!}{2(9)!} = frac{11*10*9!!}{2(9)!} = frac{11*10}{2} = 11*5 = 55 Thus even the combinations formula gives us that 1/2. Here are two blogs you may find helpful. GMAT Permutations and Combinations GMAT Math: Calculating Combinations In fact, this is really handy shortcut to know: nC2 = n(n - 1)/2 For Q2, you are 100% correct. Does all this make sense? Mike :-)

mcolbert · Answer

Can someone please explain why the answer is not B?

We know that the total number of people is 2w+1. Therefore, the number of ways to arrange the people is (2w+1)!

(2w+1)! = 3(5!)(7!), sufficient

The key here is that the question does not say that all the men are identical and all the women are identical. Why would anyone assume that?

avikroy · Answer

hello Sir,

I am having problems conceptualising the arrangement.................. Can you please help me??

the way I see it, first we can place the women ...lets say 5.... that is 5!

Then we can place atleast 1 man between 2 women............So 4 out of 6 Men can be chosen in 6C4 ways and arranged in 4! ways...

So total arrangement is 5!*6C4*4!

How to arrange the last 2 men??

Amlu95 · Answer

Its generic Ex .5 people and note it doesn't say Daniel, Mat and sandy were seated. So from this perspective, all people are identical

srik410 · Answer

This is okay to assume. In fact you have to assume everybody is different. If its same, the answer would not be C without this assumption

However this is no value of w satisfying the above equation. So this has no relevance to the standalone situation with x+1M and xW. But, with I, this makes sense and the value comes out okay

srik410 · Answer

Concise solution for C: 
Assuming (1) and (2), lets see how many ways are possible to seat x+1Men and xWomen such that no two women are sitting beside each other. 
Step1: Arrange all the men in the line and permutations -> (x+1)! ways
Step2: Choose x gaps from all the gaps possible with x+1 men in a line. This can be done x+2Cx ways.
Step3: Permutate all women within the chose gaps -> x!

So total = (x+1)!*(x+2)Cx*x!. Solving this we get x=5 so total=11

ohhno · Answer

This is a  Data Sufficiency  question.
Let:
 
 Number of women = WW 
 Number of men = MM  
Given:
M=W+1M=W+1
We need the  total number of people :
M+W=2W+1M+W=2W+1
 
Statement (1)
 
This tells us about the  arrangement , but doesn't tell us how many men and women there are.
For example:
 
 W=2, M=3W=2,\ M=3 → 5 people, possible. 
 W=3, M=4W=3,\ M=4 → 7 people, possible. 
 W=4, M=5W=4,\ M=5 → 9 people, possible.  
So there are multiple possibilities.
❌  Statement (1) alone is insufficient. 
 
Statement (2)
 
3(5!)(7!)3(5!)(7!)
This is the key.
Because men and women are distinct people, for a given M,WM,W, the number of arrangements satisfying the condition that  no two women sit together  is:
(M+1)!?(M+1)! 	ext{?}
Let's carefully derive it.
First arrange the MM men:
M!M!
This creates M+1M+1 gaps:
_M_M_M_⋯_M_\_ M \_ M \_ M \_ \cdots \_ M \_
To ensure no two women sit together, put the WW women into WW different gaps:
(M+1W)\binom{M+1}{W}
Then arrange the WW women:
W!W!
Therefore:
Number of arrangements=M!(M+1W)W!	ext{Number of arrangements} =M!\binom{M+1}{W}W!
Since M=W+1M=W+1:
=M!(W+2W)W!=M!\binom{W+2}{W}W!
Now test the possible values.
If W=5, M=6W=5,\ M=6:
6!(75)5!6!\binom{7}{5}5! =6!×21×5!=6!	imes21	imes5!
Not equal to 3(5!)(7!)3(5!)(7!).
If W=6, M=7W=6,\ M=7:
7!(86)6!7!\binom{8}{6}6! =7!×28×6!=7!	imes28	imes6!
Again, not immediately matching.
But notice:
3(5!)(7!)3(5!)(7!)
can be rewritten as:
3(5!)(7!)=3(5!)(7)(6)(5!)3(5!)(7!) =3(5!)(7)(6)(5!)
The intended arrangement formula may instead be counting  seating positions of men and women , rather than assigning individual people. Let's interpret the wording carefully.
If the question says  "number of ways of arranging the seating positions of the men and women" , it may mean arrangements of the genders only.
With M=W+1M=W+1, and no two women adjacent, the number of gender arrangements is:
(M+1W)\binom{M+1}{W}
Statement (2) as written, however, gives factorials, so it is clearly counting distinct people.
For W=5,M=6W=5,M=6, the number is:
6!(75)5!=21(6!)(5!)6!\binom{7}{5}5! =21(6!)(5!)
For W=6,M=7W=6,M=7:
7!(86)6!=28(7!)(6!)7!\binom{8}{6}6! =28(7!)(6!)
Neither equals 3(5!)(7!)3(5!)(7!).
So  there appears to be a typo/missing condition in statement (2)  as you've typed it.
If the official answer is known, send me the  screenshot of the question/options , because the expression 3(5!)(7!)3(5!)(7!) likely has a specific intended interpretation.

A few men and a few women are seated in a row. The number of men is on

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A few men and a few women are seated in a row. The number of men is on

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