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Confused about Combinatorics!

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edge Rising GMAT Star Default Avatar
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Confused about Combinatorics! Post Tue Aug 02, 2011 2:42 pm
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      1. If there are 7 people in a room, but only 3 chairs in a row, how many different seating arrangements are possible?
      2. If a group of 3 people is to be chosen from 7 people in a room, how many different groups are possible?


    What is the difference between these two questions? I am reading that the first one is about Permutations and the second one is about Combinations. Unfortunately, I am only looking for keywords (selection: order is not important => combinations // arrangement: order is important => permutations) and don't actually properly understand the concepts.

    I studied this a long time ago and would appreciate a straightforward explanation.

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    gmatboost GMAT Destroyer!
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    Post Tue Aug 02, 2011 5:44 pm
    There is nothing wrong with using the word "arrangements" to conclude that the first is a permutations question and using the word "groups" to conclude that the second is a combinations question.

    Permutations yield more results than combinations precisely because we care about the arrangements and aren't satisfied to just know the group of people selected.

    In this case, for every 1 group of people (Al, Bob, Carl), there will be 6 arrangements possible: ABC, ACB, BAC, BCA, CAB, CBA. So, the answer to #1 will be 6 times the answer to #2.

    To actually answer them:
    1. There are 7 choices for seat 1, then 6 choices for seat 2, then 5 choices for seat 3. 7*6*5 = 210.
    2. 7 choose 3 = 7!/(3!4!) = (7*6*5*4*3*2*1)/(3*2*1*4*3*2*1) = (7*6*5*4*3*2*1)/(3*2*1*4*3*2*1)
    = (7*6*5)/(3*2*1) = (7*6*5)/(6) = (7*5)/(1) = 35.

    Let me know what you think.

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    pinchharmonic Rising GMAT Star Default Avatar
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    Post Wed Aug 10, 2011 5:58 pm
    i think first you have to distinguish 3 people in seats, vs 3 people in a group. First visualize that a group of people say ABC is exactly the same as BCA. the 3 musketeers are the same group of people regardless of the order they are standing in for example. And on the other hand, 3 people seated ABC is different from BCA because, for one, C is now sitting next to A. That's the difference between order and no order. That is one hurdle. I believe just remembering things like "order matters" "order doesn't matter" will not serve you well once you hit a tough problem because they're always more cryptic than telling you order matters or not.

    Once given that you can start to use the formulas.

    I like to think of combinations as "choosing". 7 choose 3 = 7c3, and that's in fact how you can write it mathematically. Or you can say out of 7 choose 3 into a group.

    Permutations you can think of "arranging". 7 arrange 3 = 7n3. Or you can say out of 7 people let's arrange 3 people in a sequence. How many sequences would I get?

    luiscarlos59 Rising GMAT Star Default Avatar
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    Post Tue Dec 06, 2011 9:09 pm
    all the problems would contain those keywords? "arrange" , "combine" in order to identify the kind of problem?

    kul512 Rising GMAT Star Default Avatar
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    Post Thu Dec 15, 2011 1:04 pm
    Dear Edge,
    this is a common confusion and there is very common solution also.

    1. If there are 7 people in a room, but only 3 chairs in a row, how many different seating arrangements are possible?
    2. If a group of 3 people is to be chosen from 7 people in a room, how many different groups are possible?
    As i highlighted above, seating arrangement will require the permutation while group formation will not require the permutation, its only about choosing i.e. combination.

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