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.


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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21 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 !!

  6. So everytime the problem say something different it permutation? ? Wat would this be
    If andy has 7 shirts 6 pants and 8 ties how many diferent outfits can he make wiTh pants shirts and ties.
    There are 6 puppies born but the mother always feeds only 4 puppies at a time. How many differemt groups can she feed at one time?
    And are there any special words to look for in these word problems

  7. No wonder I could never figure out what I was being taught about these two. That whole order thing just confused the heck out of me. Now I get it! Thanks.

  8. Dude ur awesome I actually think I might do well on my Algebra 2 EOC that I have to take in about 45 minutes

  9. This is really helpful but there’s still two questions in my book that’s been bugging me.

    (Permutation) How many ways can a teacher select 6 students from a class of 30 to clean the room?

    (Combination) In how many ways can a principal select 6 teachers from 30 teachers to be the coordinators of 6 departments?

    • Great questions, Ariarima!

      The only way “How many ways can a teacher select 6 students from a class of 30 to clean the room?” is a permutation is if you assume those 6 students each get different jobs to clean. Like one is sweeping, one is dusting, etc. If there is no designation of jobs, then it’s a combination.

      For the second one, “In how many ways can a principal select 6 teachers from 30 teachers to be the coordinators of 6 departments?” – if you suppose that all six departments are the same, then, yes, it’s a combination. But if each department is different (like math, English, science, etc.) then it’s a permutation.

      Remember that the real use of math always has a story. Which has prompted me to write a blog post on it.

      Stay tuned…

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