Combinations vs Counting

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vijayitw: Junior | Next Rank: 30 Posts; Posts: 20; Joined: Mon May 20, 2013 12:25 pm; Followed by:1 members; GMAT Score:670

Combinations vs Counting

by vijayitw » Thu Aug 15, 2013 12:23 pm

Hello all

I am currently preparing for my GMAT in about 40 days time and am still a little confused as to when combinations should be used and when counting should be used? I would appreciate any help

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Source: Beat The GMAT — Problem Solving | Thu Aug 15, 2013 12:23 pm

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by [email protected] » Thu Aug 15, 2013 2:10 pm

Hi vijayitw,

The difference between permutation and combination questions comes down to this: Does the ORDER of things matter?

If the order matter, then you're dealing with a permutation question.

For example, if there are 5 runners in a race and there are no ties, then how many different arrangements are there for 1st through 5th place? Since we're putting the runners "in order", this is a permutation (the word "arrangement" is also a clue):

The answer would be 5 x 4 x 3 x 2 x 1 = 120 different arrangements of the 5 runners.

If order DOESN'T matter, then you're dealing with a combination question (and you need to use the combination formula)

For example, if there are 5 different colored markers, then how many different groups of two can be formed? Here, the order of markers doesn't matter, since we're forming groups (eg red-blue is the same as blue-red; it's the same group of two).

The answer would be 5!/[2!3!] = 10 different combinations of 2 markers

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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vijayitw: Junior | Next Rank: 30 Posts; Posts: 20; Joined: Mon May 20, 2013 12:25 pm; Followed by:1 members; GMAT Score:670

by vijayitw » Fri Aug 16, 2013 6:17 am

Rich

Many thanks for your advice on the same. I have a better understanding of when to use a particular method.

Regards
Vijay

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Fri Aug 16, 2013 6:48 am

If you're interested, I recently wrote an article for BTG about counting techniques (and when to use each): https://www.beatthegmat.com/mba/2013/07/ ... ons-part-i

Personally, I'm a big fan of applying the Fundamental Counting Principle (FCP) for the majority of counting questions. We have a free video on this topic: https://www.gmatprepnow.com/module/gmat-counting?id=775

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 am; Thanked: 2256 times; Followed by:1535 members; GMAT Score:800

by lunarpower » Sat Aug 17, 2013 6:02 am

maybe i'm a simpleton, but i'm surprised that i'm the only one here to interpret "counting" as referring to ... counting. like, the "one, two, three" / "make a list" kind of counting.
... which is also a perfectly respectable way to solve GMAT combinatorics problems.
in fact, just making a list and counting things can often be more efficient than any formula-based method, especially when the numbers of things involved in the problem are moderate (as they almost always will be in GMAC problems).

for instance, consider #132 in the OG quant review book (can't reproduce here).
... try 4 colors (A, B, C, D):
A
B
C
D
AB
AC
AD
BC
BD
CD
it took me less than 10 seconds to make that list (... and i'm dyslexic, so, really, yeah, it's not going to take you very long.)
that's only 10 shipping codes. not enough.
... but we're done, since we can realize (without doing the work) that adding another color will clearly create more than 2 additional possibilities. so that will give us the 12 we need.

--

the above is just an illustration of the use of LITERAL "counting", which i'd guess is what the original poster meant by "counting". (if it's not, then still note that it's an important technique.)

in terms of "when should i use this?" --
the best advice i can give is, "just try something." if it works, awesome! if not, just quit and try something else.

Ron has been teaching various standardized tests for 20 years.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Sat Aug 17, 2013 6:22 am

Hey vijayitw,

As Ron pointed out, "listing and counting" is a very valid approach. In fact, we suggest that all students take a moment to mentally list some possibilities so as to better understand the question.

If your question is, When should we consider using the "listing and counting" approach?, then there are several answers:

- if the answer choices are small numbers, then we can be certain that listing and counting won't eat up a lot of time. So, the approach may be worthwhile in these circumstances.

- if the answer choices are huge (100 or more), you might consider using some other counting techniques (e.g., Fundamental Counting Principle, combinations, etc,). HOWEVER, even if the answer choices are huge, you may find that a pattern emerges after listing a few possibilities, and you can then perform calculations based on that pattern. So, the "listing and counting" approach may even be worthwhile in these circumstances.

- One of the problems with listing and counting is that you may forget to list some of the possibilities. Also, unless all of the answer choices are very small (e.g., under 20), we can't be certain that listing and counting won't take up a lot of time. For this reason, I'd say that you should first consider other counting techniques, before committing to listing and counting.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by lunarpower » Sat Aug 17, 2013 6:54 am

Brent@GMATPrepNow wrote:unless all of the answer choices are very small (e.g., under 20), we can't be certain that listing and counting won't take up a lot of time.

well...

* doesn't have to be all of them; rather, all but one is good enough.
e.g., if the answer choices are 5, 10, 15, 20, and 1000 possibilities, then the most you'd ever have to list would be 21 possibilities (in which case the answer would default to (e)).

* ... also, even if you have more than one "big" answer choice, making the beginnings of a list can still help if you're in guessing mode (as many people are on these questions!).
e.g., if the answer choices are 10, 20, 100, 400, and 600 possibilities, then at least you can take a crack at making a list. if it's (a) or (b), then of course you'll solve the whole thing. and, if not, then as soon as you reach 21 possibilities you can at least cross off (a) and (b), thereby almost doubling your odds of guessing correctly.