# Veritas Prep Daily GMAT Challenge – Critical Reasoning and Combinatorics!

by on July 1st, 2010

Each day this week, we will be challenging the Beat The GMAT community with a Verbal and Quant question that we consider especially difficult. The first 10 people to provide the correct answer to either question in the comments section will win a Veritas Prep GMAT prep book.  Make sure to submit your answer(s) by 5pm PDT today to be eligible for the contest.  Winners will be announced in tomorrow’s Veritas Prep post.

## Combinatorics  Challenge Question

If the President and Vice President must sit next to each other in a row with 4 other members of the Board, how many different seating arrangements are possible?

(A) 120
(B) 240
(C) 300
(D) 360
(E) 720

## Critical Reasoning Challenge Question

In North America there has been an explosion of public interest in, and enjoyment of, opera over the last three decades. The evidence of this explosion is that of the 70 or so professional opera companies currently active in North America, 45 were founded over the course of the last 30 years.

The reasoning above assumes which one of the following?

(A) All of the 70 professional opera companies are commercially viable options.
(B) There were fewer than 45 professional opera companies that had been active 30 years ago and that ceased operations during the last 30 years.
(C) There has not been a corresponding increase in the number of professional companies devoted to other performing arts.
(D) The size of the average audience at performances by professional opera companies has increased over the past three decades.
(E) The 45 most recently founded opera companies were all established as a result of enthusiasm on the part of a potential audience.

## Solutions from Yesterday’s Challenge Problems

With Reading Comprehension passages, like this one that appears in the Veritas Prep Reading Comprehension book, it is important to note that the passages are often intentionally dense and technical.  Your task in reading them is not to be an expert on the subject matter, but to set yourself up to answer the questions correctly.  Those questions tend to focus much more on the author’s intent and the structure of the passage than they do on the subject matter itself.  Accordingly, as you’ll learn in the Veritas Prep Reading Comprehension lesson, your goals when reading each paragraph should be to:

1. Determine what the author is trying to accomplish
2. Reference briefly the scope of the content for future reference

Take paragraph 2, for example.  Here, the author leads by saying that the previous hypothesis worked fairly well, HOWEVER under further investigation it simply didn’t hold up.  Then, scientists adopted a second theory, called the nebular hypothesis.

If you’ve noted that much – which is made much easier by focusing on keywords like “however” that signal transition and organization – you’re in terrific shape to answer further questions.

When questions do ask about details, they’re quite often still dealing with the way that those details are organized.  Having that two-point checklist for each paragraph will be helpful for you in determining which section to revisit (the above question asks about the details of the nebular hypothesis, which we’ve noted is detailed in the second half of paragraph two).  From there, see what the question asks you to do – often times, detail-oritented questions will ask about a cause-effect relationship, in which case you need to be careful to supply the correct half (if it asks for a cause, the effect will be a tempting wrong answer choice).

This question asks for the cause of the protoplanetary disc.  Knowing that the nebular hypothesis is detailed toward the bottom of the second paragraph, you can skim for an easy-to-find word like “protoplanetary” to know where to read for that relationship. Here, it can be found in the sentence “The resulting angular momentum would have morphed the nebula into a protoplanetary disc.”  The word “resulting” in that sentence should be a nice signal for you – the angular momentum that created the disc is the result of what comes prior, so a read of the preceding sentence or two should supply the answer.  The previous sentence says that “…when a cloud reached a critical mass, it collapsed under its own gravity.”  With those two sentences, we know that a cloud collapsed under its own gravity, and the resulting angular momentum created the protoplanetary disc.  Scanning the answer choices, that is exactly what choice B says, making it the correct answer.

When approaching Reading Comprehension passages and questions, remember that your job is to approach these topics as a manager and not as a subject matter expert.  Reading for the organization of each paragraph, and for any details contained in questions, will allow you to read effectively, efficiently, and in a manner in line with how you’ll read in the business world.

Congratulations to the following people for correctly answering the question:

Vibhavender Y, Ank,  Sarathy Srinivas, Govardhan, Jeri, Pavan, Neil, Praveen, Amit, and Sachin Deshpande!  We will be sending you a message shortly requesting information about where to send your book.

Data Sufficiency Challenge Problem   Solution

Like many Data Sufficiency questions, this question, which appears in the Veritas Prep Data Sufficiency lesson book, will test your ability to consider the entire array of possible numbers, and how each might impact the answer to the overarching question (in this case, is ?).

Here, it seems logical to most examinees to take statement 1 and divide both sides by x, leaving:

x = 9

With this rephrasing of the statement (which is INCORRECT – more to come on that!), the answer to the question would be a definitive “NO”, as x is not less than or equal to zero.  As you’ll learn in the Veritas Prep Data Sufficiency book, this means that the answer would need to be A or D, given that statement 1 is sufficient (again, we’ll debunk this shortly).

Moving on to statement 2, you’ll see one of the GMAT’s greatest trickery devices at work.  First, analyze the statement, which is a clever one.  It says, conceptually, that:

Taking the absolute value of x is the same as multiplying it by negative one.

When you see a negative sign in front of a variable or a parenthetical statement (e.g. 3 – (x + 2)), you’ll be well served to insert a coefficient of 1 next to the negative for ease of understanding that step.  Here, -x may look fairly abstract (does that mean that x is negative?), but -1 (x) is clearer; you’re taking whatever x may be and multiplying it by negative one.

In this statement, then, you’ll likely note that positive numbers do not work:  the absolute value of 7, for example, is 7…which is not the same as -1(7).  But for negative numbers, it does work:  The absolute value of -7 is 7, and so is -1(-7).

Here is where you can employ a quite useful strategy, covered in the Veritas Prep Data Sufficiency lesson, called “don’t contradict yourself”.  If you’ve determined based on statement 1 that x is 9, but statement 2 tells you that it cannot be positive, you’ve done something wrong – the two statements WILL NOT contradict each other!  Now that we realize that we’ve made a mistake, we can go back to check our work, and find that:

0 satisfies both statements.  In statement 1, is the same as 9(0), and in statement 2 the absolute value of 0 is the same as multiplying 0 by -1.

Often times, Data Sufficiency questions rely on your ability to consider the numbers that don’t come to mind as often – namely negative numbers, nonintegers, and 0.  If you can keep yourself aware of the possibilities for those numbers to be in play, you’ll avoid many of the traps that the GMAT authors lay for you.

Here, let’s return to the statements, now considering 0 as an option.  In statement 1, because x could be 0, we can’t simply divide by it (dividing by 0 is undefined).  Instead, we need to take the statement:

or

Because x could be (0) and it might not be (9), the statement is insufficient.

With statement 2, we know that x cannot be positive, but it could be negative or 0.  That directly matches the question stem, so statement 2 alone is sufficient, and the correct answer is B.

Congratulations to the following people for correctly answering the question:

Ank, Sarathy Srinivas, Vibhavender Y, Jeri, Neil Zieses, Amit, Nicole, Govardhan, Pavan, and Praveen!  We will be sending you a message shortly requesting information about where to send your book.

Free Trial Class TODAY: Join Brian Galvin, Veritas Prep’s Director of Academic Programs and one of our highest-rated GMAT instructors, today Thursday July 1 for a 3-hour Combinatorics & Probability GMAT lesson.  This is lesson 12 of Veritas Prep’s 14 lesson Complete Course! RSVP TODAY.

This week only – 50% off Veritas Prep GMAT books: Just for this week, take 50% off Veritas Prep GMAT books. Use discount code VPBooks10.

• Combinatorics
option B 240

• now there are 4 member and President and Vice President must sit next to each other so let us consider them as single entity so altogetehr there are 5 member which can be aranged in 5! ways
and president and vice president can arrange themselves in 2! ways

so total number of ways 5!*2!=240

• There are 4 member and President and Vice President must sit next to each other. President and vice president can be taken as one item then total number of persons 5. Five persons can be arranged in 5 X 4 X 3 X 2 X 1 = 120 ways.

President and vice president can be sit 2 ways next to each other. Total number of ways = 120 X 2 = 240 ways

• Hi Amit,

Congratulations on correctly answering the Combinatorics and Critical Reasoning Challenge Problems correctly! Could you please send me a private message (VP_Marisa) with your shipping address and phone number? I want to make sure we send you these books as well as those that you won earlier in the week.

Thank you,
Marisa

• critical reasoning
option E

• change of answer as in e i overlooked ALL
CRitical reasoning B

• Combinatorics Challenge Question: Answer B - 240
Critical Reasoning Challenge Question: Answer B

• Hi Hardik,

Congratulations on correctly answering the Combinatorics and Critical Reasoning Challenge problems. Can you please send me a private message (VP_Marisa) with your mailing address and phone number so we can send you the books?

Thanks,
Marisa

• Combinatorics -B
Critical Reasoning - E

• Changing my answer to Critical Reasoning question - B

E is extreme: "45 most recently founded opera companies were *all* established"

Auuming "ALL" opera companies established due to public enthusiasm is not necessary.

• Hi Ank,

Congratulations on answering the Combinatorics and Critical Reasoning Reasoning questions correctly. Can you please send me a private message (VP_Marisa) with your mailing address and phone number? I'd like to send you the books from this challenge as well as those that you won earlier in the week.

Thanks,
Marisa

• Quant - B
CR - C

• Quant = B
CR = B

• Hi Pardeep,

Congratulations on correctly answering the Combinatorics and Critical Reasoning Challenge problems! Can you please send me a PM (VP_Marisa) with your shipping address and phone number? We want to make sure to send you all of the books that you've won this week.

Thank you,
Marisa Peck

• (B) FOR Combinatorics Challenge Question
(D) FOR Critical Reasoning Challenge Question

• Hi Prabhdeep,

Thanks,
Marisa Peck

• Combinatorics - B - 240
Critical Reasoning - E

• Hi Shalini,

Thank you,
Marisa Peck

• CR : A
Quant: B

• Hi M,

Thank you,
Marisa Peck

• Combinatorics Challenge Question: ANS B
since president and vice president will be together so considering them as 1 element they can be arranged in 5! ways.
They can be arranged between themselves in 2 ways
so total no. of ways = 5! x 2 = 240 so B

Critical Reasoning Challenge Question: ANS D
obviously the size of the average audience at performances by professional opera companies has increased over the past three decades and thats why company has setup more .

• Hi Shashank,

Thanks,
Marisa Peck

• Correct Option Combinatorics : 240 Option B
Correct Option Critical reasoning: Option E

• Hi Abhishek,

Thanks,
Marisa Peck

• Quant: B
CR : C

• (B) FOR Combinatorics Challenge Question
(D) FOR Critical Reasoning Challenge Question

• Combinatorics Challenge Question: B

• PS: B (simple 5! * 2!) => considering P+VP as one place
CR: C (applying negation technique: assuming that this option is talking about "professional companies")

• Correct Option Critical reasoning: E

• Combinatorics:

Consider President and VP as 1 unit and then we have 4 members, so total 5, which can be arranged in 5! ways. President & VP themselves can be arranged in 2! ways. Thus, there are 5! * 2! ways = 240.
PS: I am wondering how this can be considered a challenging problem (sorry, No offense meant).

Critical Reasoning:

Conclusion of the stem is that there has been an explosion in the public interest. If the average audience at the opera has not increased, then it nullifies the conclusion that there has been an explosion in the public interest. Thus the question stem assumes that the average audience at the opera has indeed increased over a period of 30 years.

• Forgot to mention that the answer for Combinatorics problem is B.

• For the combinatorics problem:
presindent (P) and Vice president (VP) are supposed to hang out together. so we treat them as one entity. apart from this entity, there are 4 others who need to be seated next to them. so in total we have 5 entities. thus these 5 entities can be arranged in 5! ways. now within the P and VP, they can interchange positions as VP and P, thus there are 2 ways in which P and VP can be arranged. Total number of ways- 5! *2 => 120*2 => 240 ways. Answer is B.

• for the critical reasoning:
Conclusion-> Surge of public interest in opera in the past 3 decades.
Evidence-> out of 70 operas 45 were founded in the past 30 years.

so in effect the author is trying to lay a cause and effect argument.
rise of Public interest in opera caused 45 new operas to be founded.

Now it could be that the increased founding of operas caused a rise of public interest in this field? So in effect the author must have assumed that the cause leads to the effect and not the other way around.

E very well explains that it is the public interest that caused the operas. Thus my answer is E.

• B and B

• Praveen bhai,

seems we wont bag the prize today...and the best part of challenge is "surprise" timings...

Could u explain y u picked B for CR??

• Combinatorics - Correct Answer B
---------------------------------
Number of possible arrangements (assuming President and Vice-President to be single entity) = 5!

Number of possible arrangements within President and Vice-President = 2!

Total possible number of arrangements = 5! * 2! = 240

Critical Reasoning - Correct Answer B
------------------------------------
The reasoning is a causal relationship that is as follows:

Cause: 45 of 70 currently active opera companies were founded in last 30 years.
Effect: Explosion in public interest and enjoyment of opera over last 30 years.

There is no apparent gap in reasoning, so do not waste time in pre-phrasing a supported assumption. Rather, look for a defender assumption in the answer choices.

A. Commercial viability is irrelevant. Conclusion is about increase in public interest.
B. If there had been more opera companies that had been active 30 years before, but have closed during these 30 years, then the conclusion that there had been "an explosion in public interest for opera" cannot be made. Bingo! This is the defender assumption in the argument.
C. Other performing arta are out of scope. We are talking about Opera.
D. Tricky. The increase in enthusiam may mean that size of audience has increased. But careful! This is a strengthener to the argument. Not an assumption. May be people are enjoying Opera sitting in front of TV. It would be a wrong assumption to say that the audience has increased.
E. Tricky again! This could be a strengthener. But not an assumption. May be these companies were founded because of a recent grant by the government to promote Opera.

• Regarding choice E, I just realized that E actually to be a weakener.

The reason is that the relationship stated in the argument is:

Increase in number of Opera (Cause) -> Increase in pub interest (Effect)

But choice E states that the relationship is actually reversed. Choice E states:

Increase in pub interest (Cause) -> Increase in number of Opera (Effect)

• Hi Vivek,

Thank you,
Marisa Peck

• Combinatorics: B

CR : E

• E is the best fit here. Evidence stated ti support the conclusion that Number of operas has increased over the past 3 decades.

E actually "dismiss" all other factors apart from the "enthusiasm" shown by the people.

Other factors could be govt grants, commericial benefits or tax benefits or may society development activities etc..

E exactlt brings in the fact that "all" the operas that has been recently started are "soley" becox of "people's enthusiasm.

Even though it looks like a strengther, it partly comes as a strenghtener to the evidence stated for the conclusion.

IMO E

• I am still wondering how B is the answer, that guys have selected?

Let the No. of theatres in Year 2000 = 70

As 45 companies are founded,(70-45)

Let the No. of theatres in Year 1970= 25

As per B, There were fewer than 45 professional opera companies that had been active 30 years ago and that ceased operations during the last 30 years.

"Fewer than" means it can be 1 or 44.

case 1: Let us assume 44 companies were there in 1970
And all of 44 stopped its operations in past 30 years.
So it brings us to ( 0+ 45 = 45) which is less than 70 as stated.

case 2: let us assume Minimum condition: 1 company was there in 1970

So as per B, the maximum will be 45 will be active as of 2000.But the stem says Min 70 are active..

I am not sure how I construed this math...

Plz help me out...I dont think B will be a assumption.
SO at the end of 2000 , still only 45 will be there as that 1 company closed its operation in 1970.

• Hello Govardhan,

Here is my line of reasoning.

B (assume this to be written in year 2010) says the following:
"There were fewer than 45 professional opera companies that had been active 30 years ago and that ceased operations during the last 30 years."

If we negate this statement, then B would read as follows:
"There were more than 45 professional opera companies that had been active 30 years ago and that ceased operations during the last 30 years."

Let us try to put this in a numerical example. Let us assume that there were 150 active opera companies before the year 1980. According to the argument, in 2010, there are 70 active opera companies. Out of these 70, 45 were founded between 1980 and 2010. So, we can safely say that the number of opera companies that were active before 1980 and that were closed between 1980 and 2010 = 150 + 45 - 70 = 125. In effect, what we are saying is that between 1980 and 2010, 45 new opera companies were started (which are also currently active) but 125 opera companies (which were active before) were shut down.

Now, on the basis of above calculation, can you safely conclude that public interest has increased? No. Because between 1980 and 2010 the number of opera companies that have shut down are more than those that have newly started.

So, the negation of B makes the argument fall apart.

Now, for choice E, IMO, this is in fact a weakener. The reason is that the relationship stated in the argument is:

Increase in number of Opera (Cause) -> Increase in pub interest (Effect)

But choice E states that the relationship is actually reversed. Choice E states:

Increase in public interest (Cause) -> Increase in number of Opera (Effect)

But let us wait for the official answer. May be I am making a mistake somewhere in my line of reasoning.

• Hey Govardhan,
Let me try to explain ...mathematically as you would like it......
according to the statements given....70 companies are active right now and it says 45 are opened in the last 3 decades...---->(1)
Now what B says is....
there were fewer than 45 companies before 30 years AND that ceased operations in the last 30 years...

so that means...for eg there can be 40 companies active before 30 years...
but only 15 companies might have closed in the last 30 yrs...(these 15 were active before 30 and closed in the last 30..exactly as the statement says)..(not necessarily does it mean that all the companies which were active before 30yrs were closed down in the last 30yrs...all it says is SOME of them which were active before 30 were closed in the last 30 yrs)

so that brings down the number to 25(which would be the base number for the remaining 45 companies to start in the last 3 decades) which are still active before the 30 yr period starts....now from 25 to 70...45 companies are opened in the last 30 years...so the net change is (70-40)-(40-25)=15

this net change SHOULD always be positive to indicate the increase in public interest.
(number of companies active now-no of companies initially active before the 30 yr period)-(no of companies active before 30 yr period-no of comapnies closed in the last 30 yr period)==> net increase-net decrease
so as this number (which we initially considered as 40 (active before 30 yrs)) increases say like 41,42,43,44,45,46,47.....let's take the case of 47..so the net change would be (70-47)-(47-25)= 23-22 =1 minimum value for the change to positive.

at 48, this equation fails and denotes that the public interest has actually declined or more or less remained the same and would further go down the scale if we increase the number....

so what B says is the same , but assuming a safe number of 45....it can also be said as There were fewer than 47 professional opera companies that had been active 30 years ago and that ceased operations during the last 30 years. but the author decided to be on a safe side not to consider the margins....

so for the argument to stand...it HAS to be assumed that fewer than 45 companies of WHICH were active(can be some of those 45) were closed in the last 30 yrs.

let me know if I am not clear....

• Combinatorics: B
CR: B

• VP_Marisa,
I answered correctly but I could not get price.

• Typo
I answered correctly but I could not get prize.

• critical reasoning : B

• if we try the negation test on B, then if there were more companies 30 years back and have eventually closed down, then the argument would fall apart....
let's try it on D ...the avg size of audience has not increased....yeah can be....but does that affect the argument?No...size might have remained the same and the companies might have increased( we do not need the polar opposite...only the logical opposite)
coming to E....potential audience??are we concerned about the potential or might be audience...moreover if we negate it...not all the companies were founded due to public interest....yeah MAY BE they were founded to promote some french singers...but it doesn't really affect the argument.....
A and C can be eliminated in the same way...

• Hi Vibhavender,

Thank you,
Marisa Peck

• Vibhavender,

My apologies - I already have your information. No need to resend.

Thanks,
Marisa

• Quant Perms and Combs : B

• CR: B
quant: B

• Sorry, I meant to say CR: E

• Combination -- Answer is B (240)

Make president and VP as one.. and then arrange 5 people in 5! ways and President and VP can also switch between so -- the answer is 2 * 5! = 240

• Critical Reasoning - Choice E

• Math - B

• Problem solving answer : Option B

• Combinatorics:B

Critical reasoning:B

• Comb- B
CR- B

• Hi Rosh,

Thank you,
Marisa Peck

• Irony is No American has posted the answer becox they all will be sleeping right now(early Morning) on their couch..LOL!!

• Quant : B
CR : E

• Quants:B
total no.=6;P and VP are always togather so tot people=5 and can be arranged in 5! ways;p and VP can be arranged between themselves in 2 ways. So total arrangements=2*5!=240

cr: D
negating D breaks the conclusion

• Quant: B
CR - B

• combinatorics challenge question - answer A
critical reasoning challenge question- answer E

• Math - B (5!2!)
CR - E

In North America there has been an explosion of public interest in, and enjoyment of, opera over the last three decades. The evidence of this explosion is that of the 70 or so professional opera companies currently active in North America, 45 were founded over the course of the last 30 years.

Evidence: 45 founded in last 30 yrs; 70 operating now
Conclusion: Substantial increase in Public Interest in opera
Assumption: will connect 45 companies with public interest ie it assumes that 45 were founded bcoz of public intertest.

The reasoning above assumes which one of the following?

(A) All of the 70 professional opera companies are commercially viable options. (commercial viability not directly discussed, but if at all, its all ready mentioned that currently there are working)

(B) There were fewer than 45 professional opera companies that had been active 30 years ago and that ceased operations during the last 30 years. (if some were closed and some new opened...thn it will weaken)

(C) There has not been a corresponding increase in the number of professional companies devoted to other performing arts. (other art out of scope)

(D) The size of the average audience at performances by professional opera companies has increased over the past three decades. (avg audience size out of scope)

(E) The 45 most recently founded opera companies were all established as a result of enthusiasm on the part of a potential audience. (if 45 companies were established only bcoz of interest, it will eliminate any alternate possibility of creation of these companies)

• prashant bhai,

u r in my Wing!!

combinatorics challenge question - B
critical reasoning challenge question - E

• Quant-B
CR-D

• My bad on CR-D pick. Sorry!!!

CR-B
(A) This choice is about "effect of the effect" i.e., All 70 POCs are viable options.
(C) This answer is talking about companies devoted to other arts. Out of scope.
(D) This choice is about "after cause of cause", the audience size has increased.
(E) This choice claims that "effect is the cause of the claimed cause." Reversed the cause-effect relationship.

2x5!=240

combinatorics challenge question - B
critical reasoning challenge question - B

• Combinatorics : B
CR: E

• Combinatorics : B
CR: D

• Combinatorics : B
CR : E

• for math problem. I choose B, 240

for CR , I choose D

• B & B

• PS: B
5!*2! = 120*2 = 240

CR: E

A - Out of scope
B - Inference not assumption
C - Out of scope
D - Inference not assumption
E - Best Choice

• Comb. Answser- B

CR answer- E. It makes the most sense.

• Quants: B
Verbal: E

• 1) 5! 2!=240
Ans. B
2) we need to find why the public interest exploded in the last 3 decades. It is given in B.
Ans. B

• Critical reasoning answer : D

shiva on July 1st, 2010 at 3:02 pm

There are 4 member and President and Vice President must sit next to each other. President and vice president can be taken as one item then total number of persons 5. Five persons can be arranged in 5 X 4 X 3 X 2 X 1 = 120 ways.

President and vice president can be sit 2 ways next to each other. Total number of ways = 120 X 2 = 240 ways

• CR: B
A and E use "all" which is not necessarily assumed. C is irrelevant.

• combinatorics:B
CR:E

• Answer to the quant question is: B
Answer to the CR question is: E

• Quant answer is 5! *2! =240 ...... (B)
Critical Reasoning ...................... (E)

• Sorry I was out of station....My answers are:
Combinatorics: B
CR : E