GMAT question - symantic problem?

[Misfire]

This was in a GMAT book, and I think the wording is a little ambiguous. I can see how to get two answers depending on how I read it. Thoughts?

Entries in a particular lottery game are made up of three digits, each 0 through 9. If the order of digits in the entries matters, how many different possible entries exist in which all three digits are not equal?

[ ]516
[ ]720
[ ]989
[ ]990
[ ]1321

[madnak]

I get 720, and it doesn't seem close. The first digit is one of the ten, the second number can be any of the other 9, and the third can be any of the remaining 8. 10*9*8=720. Where is the ambiguity you're seeing?

[Sephus]

the question could mean "how many different possible entries exist in which it's not the case that all three digits are equal?"

like 001. you could say "are all three digits equal? no. therefore, all three digits are not equal."

in this case, you get 990.

[Misfire]

That's what I got too, but the answer explanation says 990. There are 1000 ways to pick numbers, but 10 of them have all three with the same number (1 1 1, 2 2 2, etc).

I would buy this if the question read, "in which not all three digits are equal" rather than "in which all three digits are not equal."

[madnak]

Oh, another way to prove it is to look at how many repeats there are for each digit.

So for 1, there's 011, 111, 211, 311... and there's 110, 111, 112, 113... and there's 101, 111, 121, 131... There are ten numbers in each group. 30 numbers altogether. But 111 appears in all three groups - we don't want to count it three times, only once, so we really have 28 unique numbers.

Since there are only 3 digits in each number, only one of the digits can repeat. Therefore, every digit will repeat the same number of times through the whole 1000 possible 3-digit numbers. We know this number is 28, and there are 10 digits, so the total number of repeats is 280. There are 1000 numbers altogether, so there are 1000-280=720 numbers that don't repeat.

[madnak]

[ QUOTE ]
That's what I got too, but the answer explanation says 990. There are 1000 ways to pick numbers, but 10 of them have all three with the same number (1 1 1, 2 2 2, etc).

I would buy this if the question read, "in which not all three digits are equal" rather than "in which all three digits are not equal."

[/ QUOTE ]

Effing dumb. [censored] question, period. What kind of lame question would that be, even if it were worded properly?

(All three digits are not) (equal) is clumsy language. (All three digits are) (not equal) isn't great either, but is the better interpretation IMO. Just poorly written. Ass.

[Misfire]

Those were my thoughts. Just wanted to make sure I wasn't just misreading.

[Phil153]

"in which there are at least two unique digits"

How effin hard is that? The way the question is written 720 is more correct than 990 imo.

[mickeyg13]

I agree it's poorly worded, but it's incorrect to use the 720 interpretation. The question said nothing about pairs of digits, so you have no reason to disallow pairs of digits.

[Fly]

[ QUOTE ]

If the order of digits in the entries matters, how many different possible entries exist in which all three digits are different?


[/ QUOTE ]

This is what they would have asked if they were looking for 10*9*8.

[madnak]

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I agree it's poorly worded, but it's incorrect to use the 720 interpretation. The question said nothing about pairs of digits, so you have no reason to disallow pairs of digits.

[/ QUOTE ]

The question implies that all three digits are unequal. This disallows pairs.

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If the order of digits in the entries matters, how many different possible entries exist in which all three digits are different?


[/ QUOTE ]

This is what they would have asked if they were looking for 10*9*8.

[/ QUOTE ]

"Different" and "not equal" are close enough.

[Philo]

[ QUOTE ]
This was in a GMAT book, and I think the wording is a little ambiguous. I can see how to get two answers depending on how I read it. Thoughts?

Entries in a particular lottery game are made up of three digits, each 0 through 9. If the order of digits in the entries matters, how many different possible entries exist in which all three digits are not equal?

[ ]516
[ ]720
[ ]989
[ ]990
[ ]1321

[/ QUOTE ]

Yes, the question is ambiguous. The ambiguity is in the scope of the negation (called a 'scope ambiguity'), which is a common error for students learning logic. The sentence "How many different possible entries exist in which all three digits are not equal?" is ambiguous between (1) How many different possible entries exist in which it is not the case that all three digits are the same digit?; and (2) How many different possible entries exist in which it is not the case that any two digits are the same digit? If you read the question the first way the answer is 990, while if you read the the second way the answer is 720.

Unfortunately many standardized tests are poorly written. The GRE, the LSAT, the GMAT, all often contain poorly written questions. For those who can spot the ambiguities and otherwise poorly written questions the only way to get good at taking these tests is to develop a feel for which question is actually being intended. Many IQ tests are also very poorly written.

I once took a multiple choice exam in a psychology class where one question had one answer choice that was what our textbook said was the right answer, and another answer choice that was the actual correct answer. Knowing the textbook was wrong I chose the actual correct answer, and when it was marked as wrong when I got the test back I asked the professor why, pointing out that the textbook was wrong. His reply was, 'I know, but I was testing you on the book."

[stinkypete]

imo 990 is the only correct way to interpret this, but i agree that the wording is stupid if that's not what they're trying to screw you on - since this is on the quant part, the wording should be more clear.

the test will likely not be that retarded. test prep book makers are just out to make some monies and the quality control can be pretty atrocious.

[Sephus]

[ QUOTE ]
Knowing the textbook was wrong I chose the actual correct answer, and when it was marked as wrong when I got the test back I asked the professor why, pointing out that the textbook was wrong. His reply was, 'I know, but I was testing you on the book."

[/ QUOTE ]

the play here is to punch him in the face.

[madnak]

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imo 990 is the only correct way to interpret this

[/ QUOTE ]

The 720 grammar is perfectly correct and less convoluted.

[luckyme]

[ QUOTE ]
The sentence "How many different possible entries exist in which all three digits are not equal?" is ambiguous between (1) How many different possible entries exist in which it is not the case that all three digits are the same digit?; and (2) How many different possible entries exist in which it is not the case that any two digits are the same digit?

[/ QUOTE ]

I read the question as "How many different possible entries exist in which all three digits ARE equal?"
Then I took the the 'not' as meaning "all the other cases".
Obviously too simplistic, but I can't quite figure out why.

luckyme

[madnak]

The problem here is the adverb "not." It can modify either the verb "are" or the adjective "equal." Logically the former case gets you "the digits not (are equal)," the latter gets you "the digits are (not equal)." This is like the "spinning dancer" problem, people interpret it differently.

In the 990 case "not" and "equal" both modify "are." When "not" modifies the main verb, we can usually apply it to the whole sentence. The clause in this case. Applying "not" to the clause yields "it's not the case that the digits are equal." In the 770 case it's easy to avoid the adverb - substitute "not equal" for "unequal" and we get "the digits are unequal."

("All three" is redundant - I leave it out because the writer should have done the same.)

These situations are one reason why good writers avoid adverbs whenever possible. It also helps explain some stylistic conventions - "cannot" is clearer than "can not" because it prevents this sort of ambiguity.

[stinkypete]

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[ QUOTE ]
imo 990 is the only correct way to interpret this

[/ QUOTE ]

The 720 grammar is perfectly correct and less convoluted.

[/ QUOTE ]

i'm not gonna argue about correct but it's definitely not less convoluted imo.

[madnak]

Eh, convolution is subjective. But all I have to do is change "not equal" to "unequal" and I've fixed the 720 version. Can you fix the 990 version without restructuring the sentence?

"Aren't" instead of "are not" makes the 990 interpretation more intuitive, but doesn't solve the problem. I can't think of any other simple fixes.

[Philo]

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imo 990 is the only correct way to interpret this,

[/ QUOTE ]

If the question is ambiguous there cannot be only one correct way to interpret it.

[stinkypete]

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imo 990 is the only correct way to interpret this,

[/ QUOTE ]

If the question is ambiguous there cannot be only one correct way to interpret it.

[/ QUOTE ]

and?

[willie24]

[ QUOTE ]
This was in a GMAT book, and I think the wording is a little ambiguous. I can see how to get two answers depending on how I read it. Thoughts?

Entries in a particular lottery game are made up of three digits, each 0 through 9. If the order of digits in the entries matters, how many different possible entries exist in which all three digits are not equal?


[/ QUOTE ]

if 720 is right, the question should read: "...in which none of the 3 digits are equal"

if 990 is right (which is the way i read it) it should be something like "...excluding those where all 3 digits are equal"

despite the ambiguity, it seems to me that the reader should be able to figure out that the writer meant 990. it would be an awfully stupid way to write the 720 question...in other words, if the writer meant 720, he would have to struggle to arrive at such an odd wording. for 990, i think the words could easily come out how they did- and the writer might not notice the ambiguity, since his grammer can be correct for what he means.

[Philo]

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imo 990 is the only correct way to interpret this,

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If the question is ambiguous there cannot be only one correct way to interpret it.

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and?

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And the question is ambiguous would probably be the next line.

[madnak]

[ QUOTE ]
despite the ambiguity, it seems to me that the reader should be able to figure out that the writer meant 990. it would be an awfully stupid way to write the 720 question...in other words, if the writer meant 720, he would have to struggle to arrive at such an odd wording. for 990, i think the words could easily come out how they did- and the writer might not notice the ambiguity, since his grammer can be correct for what he means.

[/ QUOTE ]

I disagree, and on the contrary I can't see how anyone could write the 990 question like that. But there's enough dissent here that the interpretations are probably equivalent.

I still think the 720 question makes more sense, because the 990 question is inane. I mean this is the GMAT, not Kindergarten.

[willie24]

when i try to read the question with the 720 meaning in mind...

"If the order of digits in the entries matters, how many different possible entries exist in which all three digits are not equal?"

i find myself asking the writer - equal to what?

[madnak]

That's implied - the other digits. It's identical in both interpretations.

[willie24]

im not so sure it's identical.

the subject is: each of the digits within the group of digits
they can either "be equal" or "not be equal." the idea of them "being not equal" (trying to mean that each is unique) is extremely awkward to me for some reason.

maybe it's because for each to be unique, inequality has to be established between digit A and B, between digit A and C and between digit B and C. inequality (meaning uniqueness) throughout the group must be established through inequality of each pair - and relationships between individual pairs is NOT implied in the sentence.

for instance, to solve this issue, you would say: "...in which each of the 3 digits is not equal to either of the other 2."

im not 100% positive that i'm right, but that's the hang up for me with the 720 meaning

[willie24]

in other words, i guess im saying that 3 digits cannot be unequal. only 2 of anything can be unequal.

on the other hand, an infinite number of digits can be equal or not be equal.

[madnak]

[ QUOTE ]
maybe it's because for each to be unique, inequality has to be established between digit A and B, between digit A and C and between digit B and C. inequality (meaning uniqueness) throughout the group must be established through inequality of each pair - and relationships between individual pairs is NOT implied in the sentence.

[/ QUOTE ]

If you're breaking it down this way, it's easy to put it the same way wrt "equal." Equality has to be established between digits A and B, between digits A and C, and between digits B and C.

If digit C is equal to the other digits, then it must be equal to A, it must be equal to B, etc. But the pairing seems pathological. It seems like you're using pairing because there are three digits and you're stuck on the "one digit in, two digits out" mode of thinking. So just imagine it with 5 digits instead to resolve that issue.

"All five of the digits are equal" and "all five of the digits are unique" describe similar properties. Both statements indicate a relationship each digit has with the other digits. Both statements must apply without exception (in the former case, if any two of the digits are different the statement is false, and in the latter case, if any two of the digits are the same the statement is false).

I can see one logical difference (maybe this is what you're getting at) - it's easier to evaluate the claim of the first statement because the relations are transitive, while the relations in the second statement are intransitive. That is, if a=b and b=c, then a=c. But if a!=b and b!=c, that doesn't mean that a!=c. But this is a logical difference - it doesn't affect the grammar. I can see it playing an intuitive role - fair enough. But my intuition doesn't match yours. It's hard for me to read this in the 990 way. The 720 interpretation seems simpler and more obvious.

[madnak]

[ QUOTE ]
in other words, i guess im saying that 3 digits cannot be unequal. only 2 of anything can be unequal.

[/ QUOTE ]

Logically I see equality as a binary relation. I admit that I could be wrong.

If you're right, then the 990 statement may work better. Maybe "equal" can apply to a collective noun, while "unequal" can't. That seems strange to me, but it's possible.

Can someone clarify these points?

[CallMeIshmael]

[ QUOTE ]
the question could mean "how many different possible entries exist in which it's not the case that all three digits are equal?"

like 001. you could say "are all three digits equal? no. therefore, all three digits are not equal."

in this case, you get 990.

[/ QUOTE ]


obv a lot of v smart people are disagreeing here, but this is, imo, the correct way to look at it.

[stinkypete]

[ QUOTE ]
[ QUOTE ]
the question could mean "how many different possible entries exist in which it's not the case that all three digits are equal?"

like 001. you could say "are all three digits equal? no. therefore, all three digits are not equal."

in this case, you get 990.

[/ QUOTE ]


obv a lot of v smart people are disagreeing here, but this is, imo, the correct way to look at it.

[/ QUOTE ]

i don't think any smart people are saying the 720 interpretation makes more sense

[willie24]

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I can see one logical difference (maybe this is what you're getting at) - it's easier to evaluate the claim of the first statement because the relations are transitive, while the relations in the second statement are intransitive. That is, if a=b and b=c, then a=c. But if a!=b and b!=c, that doesn't mean that a!=c.

[/ QUOTE ]

yes this is exactly what i was trying to get at
i don't follow the difference between groups of 3 and 5 or what you were trying to say with that. probably beside the point anyway.

[The Dude]

It actually took me reading other people's responses to be sure where the ambiguity was, because it was pretty clear to me that 990 is what they were looking for.

[CallMeIshmael]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
the question could mean "how many different possible entries exist in which it's not the case that all three digits are equal?"

like 001. you could say "are all three digits equal? no. therefore, all three digits are not equal."

in this case, you get 990.

[/ QUOTE ]


obv a lot of v smart people are disagreeing here, but this is, imo, the correct way to look at it.

[/ QUOTE ]

i don't think any smart people are saying the 720 interpretation makes more sense

[/ QUOTE ]


If I had to bet on some sort of general intelligence contest, I dont think there is a person who has posted in this thread that Id rather have than Madnak. Except maybe Sephus, not sure, it would be close.

(obv disclaimer re: sample size of posts, etc)

[Sephus]

madnak is smarter than me.

[CallMeIshmael]

thats why it was only a maybe!

[Philo]

[ QUOTE ]


despite the ambiguity, it seems to me that the reader should be able to figure out that the writer meant 990. it would be an awfully stupid way to write the 720 question...in other words, if the writer meant 720, he would have to struggle to arrive at such an odd wording.

[/ QUOTE ]

Scope ambiguities are pretty common. Here it's the difference between "all three digits are not equal" and "not all three digits are equal." I have no problem reading the first one as saying that no two digits are the same digit (i.e., it is true of all three digits that none of them is the same as any other digit), and the second one as saying that not all three digits are the same digit.

[Phil153]

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the question could mean "how many different possible entries exist in which it's not the case that all three digits are equal?"

like 001. you could say "are all three digits equal? no. therefore, all three digits are not equal."

in this case, you get 990.

[/ QUOTE ]


obv a lot of v smart people are disagreeing here, but this is, imo, the correct way to look at it.

[/ QUOTE ]

i don't think any smart people are saying the 720 interpretation makes more sense

[/ QUOTE ]
I would take madnak over you any day.

What's more, he actually provided some linguistic analysis instead of "I'm a smart dude and therefore my opinion is right". If you actually were intelligent, you'd realize that.

Can you refute his linguistic analysis? All you've done so far is blow hot air.

[stinkypete]

phil,

fix pokerev and i'll refute whatever analysis you want me to refute

[stinkypete]

i just don't see how anyone could think that using the words "not equal" to mean "unique" when 3 digits are in question is what was intended.

[borisp]

[ QUOTE ]
I read the question as "How many different possible entries exist in which all three digits ARE equal?"
Then I took the the 'not' as meaning "all the other cases".
Obviously too simplistic, but I can't quite figure out why.

luckyme

[/ QUOTE ]
This is the "correct" way to think about it, coming from a person that has taught these standardized tests. It is probably what the writer was thinking.

A simple technique for SAT, LSAT, GMAT, etc., is that whenever a problem has a negation, remove the negation, and then evaluate the content. The negation means everything else.

[MC Chris]

didn't read any posts so far, but the answer is 990 and the question is perfectly clear.

[Philo]

[ QUOTE ]
i just don't see how anyone could think that using the words "not equal" to mean "unique" when 3 digits are in question is what was intended.

[/ QUOTE ]

Consider the number 788. Now ask the question, "are all three digits not equal?" That is to say, are all three digits unequal? The first digit is not equal to either of the other two, so that digit has no equal, but this is not true of the second or the third digit. Therefore it is not the case that all three digits are unequal (since two of them are). I don't think there's anything unreasonable in parsing the sentence this way, and again, this type of scope ambiguity is quite common. The scope of the negation is ambiguous between "all three are not equal" and "not all three are equal."

[madnak]

[ QUOTE ]
This is the "correct" way to think about it, coming from a person that has taught these standardized tests. It is probably what the writer was thinking.

A simple technique for SAT, LSAT, GMAT, etc., is that whenever a problem has a negation, remove the negation, and then evaluate the content. The negation means everything else.

[/ QUOTE ]

The majority of the posters here think the 990 interpretation is more intuitive, so I'm willing to concede that point.

But it doesn't matter; even a .1% rate of error due to sloppy writing should be considered unacceptable. Test questions should be written for clarity, and semantic ambiguity should never be tolerated. The test results affect people's lives, and it pisses me off that bad writing can skew them.

[Philo]

[ QUOTE ]
[ QUOTE ]
This is the "correct" way to think about it, coming from a person that has taught these standardized tests. It is probably what the writer was thinking.

A simple technique for SAT, LSAT, GMAT, etc., is that whenever a problem has a negation, remove the negation, and then evaluate the content. The negation means everything else.

[/ QUOTE ]

The majority of the posters here think the 990 interpretation is more intuitive, so I'm willing to concede that point.

But it doesn't matter; even a .1% rate of error due to sloppy writing should be considered unacceptable. Test questions should be written for clarity, and semantic ambiguity should never be tolerated. The test results affect people's lives, and it pisses me off that bad writing can skew them.

[/ QUOTE ]

I agree that the more common reading would be the one with the answer 990. But the question still contains a scope ambiguity. Even students of logic have a hard time discerning ambiguous questions involving scope ambiguities like this, because they are used to interpreting the scope of the negation just one way. Unfortunately these types of poorly written questions do appear too often on standardized tests like the GRE, LSAT, and GMAT.

Unless I've missed it in the thread somewhere I haven't seen anyone explain what they think is wrong with the second reading where the negation scopes over just the equality, as in "all three digits are not equal," which is the actual wording of the question, and which translates as "for all x, if x is a digit, then x is not equal to any other digit."

[borisp]

I guess a reasonable response to this is that one must recognize a bit of metagame with regard to these standardized tests. But I agree that the wording is ambiguous, and either interpretation is valid, outside of this scope.

[vulturesrow]

Wow you can tell this forum definitely leans in the direction of math guys, cant believe there is 46 posts on this. I thinks it patently obvious that it should be read in the manner that gives you 720.

Question metagame analysis : 990 is way too easy to come up with. /images/graemlins/wink.gif

PS FWIW I scored a perfect on the reading comprehension portion of the ACT /images/graemlins/wink.gif

[Philo]

[ QUOTE ]
That's what I got too, but the answer explanation says 990. There are 1000 ways to pick numbers, but 10 of them have all three with the same number (1 1 1, 2 2 2, etc).

I would buy this if the question read, "in which not all three digits are equal" rather than "in which all three digits are not equal."

[/ QUOTE ]

Your analysis is correct. The first way of phrasing it is unambiguous since the negation scopes over the universal quantifier 'all'. The second way of phrasing it, which was the way it was phrased in the original question, is ambiguous since it does not make clear what the scope of the negation is supposed to be.

[willie24]

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PS FWIW I scored a perfect on the reading comprehension portion of the ACT

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my dick is 8 inches

[Philo]

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my dick is 8 inches

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I wouldn't worry about it too much, I'm sure you have other redeeming qualities.

[willie24]

touche