I share a twin house with my neighbors (i.e. we are attached) and we like to decorate our front porches for the holidays the same, so that our ‘house’ is coordinated. Every year we do something different, and this year we decided to hang holiday ornaments along with the lights – so a variety of Christmas balls and various large ornaments hanging from the porch. As we were trying to decide the most ‘pleasing’ order to hang these, I realized we were basically discussing combinations and permutations, which naturally got me thinking about working with this in math class.
Permutations and combinations are often very confusing for students. Basically you have a group of things (numbers, objects) and you are going to pick a certain amount from that group of things, and depending on whether order matters, you either have a certain number of combinations of things you can make or a certain number of permutations. Combinations are the possibilities of things chosen when order doesn’t matter. Permutations are the possibilities of things chosen when order does matter.
As an example:
We have 3 Christmas Balls – green, red, blue. If I want to choose two, order doesn’t matter, than it’s a combination, so how many combinations will I get?
3 combinations: Green, Red; Green, Blue; or Red, Blue
But, if order does matter, then we have a permutation, so how many permutations are there?
6 permutations: Green, Red; Red, Green; Green, Blue; Blue Green; Red, Blue; Blue, Red.
Now, there’s also the whole idea of replacement and no replacement, but I am not going to get into that here. Working with students, you would want to start with small numbers of objects so they can create the combinations and permutations by hand. But then, you’d want to lead into more complicated things such as lottery numbers and chances of winning, where finding all the combinations and/or permutations is hard to do by hand, thus requiring a formula to make it more efficient, and then eventually, if you really want to do comparisons and have interesting discussions about many real-world examples, you’d want to incorporate technology to help be even more efficient. Here’s a nice page I found that discusses the differences between combinations and permutations and the different formulas needed and provides some good examples.
Below is how you can calculate permutations and combinations when you know your sample size (n = number of things you have) and how many you are choosing (r) from that group of n things. This video shows how to do this on both the Casio Prizm and the TI-84+CE.
Clearly in my front-porch, neighbor decision making, order actually mattered. We wanted a pleasing arrangement. We therefore were looking for permutations – how to choose six balls from a possible 10, so 10P6. There are a staggering number of permutations – 151,200. Who knew holiday decorating had so many choices!!! Needless to say we did not try to look at all of them – but good to know we have so many options for the years to come!