Manhattan Practice Test 4-2 #7 and #18

[dans.other.email.address]

Questions 7 and 18 in the second quant section of practice test 4 are both combination problems.

Number 7 contains the following wording: "The number of ways to pick 3 of the 5 cards such that card number 1 is included."

Number 18 contains the following sording: "Which of the following could represent the total number of ways to select the two cards..."

The answer and explanation to both questions made it clear that combinations, not permutations, were being tested. I read both as permutation questions, because "total number of ways" does not indicate that sequence is irrelevant, but seems to emphasize total discrete arrangements.

My understanding has been that wording such as "total numbers of ways," without further qualifications, indicates that different arrangements qualify as different "ways," and therefore indicates a permutation problem. I thought that "distinct groups of cards" or "different sets of cards" or something like that would indicate combinations instead.

I grasp the concept. I thought I could interpret the wording as well, but your practice test makes me doubt that.

If this was a lower-quality practice test (you know, certain "other brands"), I would suspect that the test was written wrongly, but it's hard to imagine Manhattan making a mistake like that. So am I misunderstanding how these questions are worded on the GRE?

[tommywallach]

Hey There,

When trying to differentiate between combinations and permutations, I'd recommend you don't become attached to any words or phrases signalling what to do. Instead, you should think about the underlying concept: does order matter or not? If order matters, it's permutation. If order doesn't matter, it's combination.

I'd have to see the actual questions to tell you further (our tests are adaptive, like the real test, so there is no consistent #7 or #18). However, "total number of ways" sounds like a legitimate way to introduce EITHER of the two. The word ARRANGEMENT signals order matters, the word GROUP tends to signal order doesn't matter. But "total number of ways" is just both. "ways" is DEFINITELY not a synonym for "arrangement":

1. We are making a team of 6 people from a pool of 9. How many different ways can the team be formed? (order doesn't matter)

2. Jennifer has to take 4 meetings tomorrow: one in the morning, one in the afternoon, one in the early evening, and one at night. If there are 10 possible people she could meet with, how many ways could she arrange her schedule? (order does matter)

3. A team is made up of 1 goalie, 2 defenders, 2 midfielders, and 1 forward. If there are 10 people who could be on the team, and we allow that the same people playing different positions would make for a different team, how many ways can we construct the team? (order both does and doesn't matter).

Make sense?

-t

[dans.other.email.address]

Mr. Wallach,

Thanks for your reply. Your explanation certainly makes sense.

For what it's worth (& for future readers), here are the problems in full:

First problem:
A set of cards is numbered 1 through 5.

Quantity A: The number of ways to pick 3 of the 5 cards such that card number 1 is included

Quantity B: The number of ways to pick 3 of the 5 cards such that card number 1 is excluded

Here, the wording seems a little ambiguous to me. Obviously, it makes more real-life sense to be interested in the different card combinations, but a scenario in which card sequence matters also seems conceivable to me. That may be my error in understanding.


Second problem:
Joe has more than 4 and fewer than 8 classic baseball cards. He wants to auction off two of them for charity. Which of the following could represent the total number of ways to select the two cards to auction, depending on the number of cards Joe has?
Indicate all that apply.

Here a combination definitely makes far more sense.

[tommywallach]

Hey Dan,

Thanks for the posting. Remember that permutations are about ORDER mattering. You're right that USUALLY order matters when you deal with numbers (123 is different from 321 or 231). That's why you need to read the questions carefully. This one simply wants to know how many ways you could pick up the cards and end up with the 1. In that example, the grabbing of the three cards pretty much happens at the same time: 123 = 321 = 231, because you end up with the same three cards no matter what.

As you said, the second problem is a little more obvious, as Michael Jordan-Magic Johnson would clearly be the same card set as Magic Johnson-Michael Jordan.

Thanks!

-t