Permutations and Combinations – How to Tell the Difference

Have you ever taught permutations and combinations?

Do you use the words, “In permutations, order matters”?

There are lots of permutation problems where there’s no ordering whatsoever. But they are still permutations. So telling the difference between a combination and permutation can be difficult if you use the ol’ “order matters” rule. Like this:

Notice there was no “order” in the permutation – each cat is having something different done with it. (And woe for the cat who’s being eaten.)

The real differentiating factor between permutations and combinations is this:

If the things being chosen are going to do (or have done to them) the same thing, it’s a combination.

and

If the things being chosen will do (or have done to them) different things, it’s permutation.

Here’s another example, along with a tip to choosing which to use:

Will this help your students? And check out the next post on how to do the calculations for these.



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15 Responses to Permutations and Combinations – How to Tell the Difference

  1. Bon – It’s a much better idea than the “order matters” thing. I use this with my students and it works. There are also problems on the SAT where this will be useful. It was something like: How many ways are there to choose a President, Vice-President, and a Secretary from five people. No order there, but it’s a permutation problem! A slightly more complicated way to do this is to first calculate how many ways to select the three people (a combination), then ask how many ways can you assign the three chosen people to fill the three offices. Then you multiply the two answers. However, your way is much easier. Thanks for the tip.

  2. Thanks so much, Allen. It is frustrating to students to try to figure out if where there is “order” when there is, technically, none.

    When I create my own questions, I always make sure to match the permutation version with the combination version. On a recent test, I asked students to determine which to use between these:

    A. From five paintings, you need to choose three for your new office.

    B. From five paintings, you need to choose one for the lobby, one for the bathroom, and one for the conference room.

    I think when students see them back to back like this, it’s much more helpful to learn the differences.

    Thanks for stopping by!

  3. I’m still confused. I have some math problems, and book has this under the permutation section, but I can’t see why.
    18. Fifteen students ask to visit the admissions representative from State University. Each visit includes one student. In how many ways can ten
    time slots be assigned?
    19. How many different nine-player batting orders can be chosen from a baseball squad of 16?
    20. The prom committee has four sites available for the banquet and three sites for
    the dance. How many arrangements are possible for the banquet and dance?

    • Mike – notice the time slots are all different. When you have each chosen thing (the students) doing different things (going at different times), then it’s a permutation.

      For #19 – when you bat makes a difference. So if you are first at bat or third, that’s different. Different things for each chosen player to do means permutation.

      And #20 – This one goes back to the multiplication principal. 4 x 3 would give you the various arrangements. If you notice, textbooks often throw in something from previous sections to help you make the connections that different methods will give you same results.

      Thanks for your question, Mike!

  4. Wow– this makes so much more sense than the “order matters” deal. Wish my finite professor had taught us this way!

  5. Thank you very much Bon! This made me understand the difference much better, instead of just using the textbooks. I will suggest these videos to my friends and teachers !!

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