October 18th | Order your combinations
Dear Mathagony Aunt
My students find it difficult to remember the difference between a permutation and a combination. They understand that either "order matters" or "order doesn't matter". Have you any ideas that might help them remember which is which?
Dear Reader
You are writing to someone who has always had to remind herself whether I am looking at a problem involving permutation or combination! Maybe, I have always been too lazy to try and create a way for me to remember. Here, I focused on why I found it so difficult to differentiate between them using the words from the definitions.
The Collins Concise English Dictionary gives the definition of permutation as
"an ordered arrangement of the numbers, terms, etc., of a set into specified groups: the permutations of a, b, and c, taken two at a time, are ab, ba, ac ca, bc, cb."
And the definition of combination as
"an arrangement of the numbers, terms, etc., of a set into specified groups without regard to order in the group." UGGHHH!
Your letter sent me on a search through piles of textbooks, and a good couple of hours on the internet. I could find no reference for any particular trick of remembering the difference. Most definitions pointed out that repetition of items was not allowed for either a permutation or combination and that order does not matter for a combination and order matters for a permutation. The language left me confused! However, I did find out from one American textbook, ‘College Algebra with Trigonometry’, that there is a name given to arrangements where repetition is allowed. When order matters with repetition it is called sample, when order doesn't matter this is called a selection.
I have illustrated my confusion in the cartoon, which I explain a little here. I considered the arrangements of the words, "order doesn't matter". Although there are six different arrangements, all count as the same combination, thus there is only one combination from these three words (even though there are six permutations. As order matters each different arrangement counts as a separate permutation). For the two words "order matters" there are two permutations: 'order matters' and 'matters order'.
But I did eventually find a way to help me remember and I now share this with you below.
In the National Lottery you choose six numbered balls out of a set of 49 (ignoring the bonus ball). The numbers you choose matters, not the order in which you choose them. You don't have to guess the order in which the balls come out of the tumbler, "the order doesn’t matter". 2, 43, 44, 1, 23, 13 and 23, 43, 1, 44, 13, 2 are the same combination. Thus your choice of numbers is a combination. In the case of National Lottery the number of combinations is found by working the number of ways six numbers can be chosen out of 49. In mathematical notation this is usually written as 49C6 or as C(49, 6). For any reader that didn’t realise, there is a button on the calculator that you can use to work this out! Otherwise the number of combinations is calculated as 49! ÷ ((49 – 6)! × 6!) = 49! ÷ (43! × 6!) = 13983816.
Looking at the word "permutations" I thought of "permutations" and genetics. Our individual genetic code is locked away in our DNA. If you like, our DNA is a blueprint of us as individuals. The order of the code in the DNA is important. Without going into detail, we could look at one small part of the DNA and see the following sequence (of nucleosides), represented here as letters: G, C, A. If this changes, or mutates, we have changed the code: we have a MUTATION. Suppose we swap the last two letters, we get G, A, C. This could have profound implications on what happens to us. Indeed mutation is what is thought to lead to evolution and cancer. So the effect can be beneficial or harmful, but what is important is that the order is important in "permutations"! The various arrangements are different permutations, or in the case of the genetic code they are different mutations.
In genetics (tRNA) we can choose three nucleosides out of four different nucleosides to form codons. These codons are used to pick one of the 20 amino acids, which go to make up proteins. We can represent the calculation as 4P3 = P (4, 3) = 4! ÷ (4 – 3)! = 4! ÷ 1! = 4! = 24.
I hope you find that this aids the memory. Making me focus has certainly helped me to remember the difference, thank you!
An permutation and combination generator can be found at www.wcrl.ars.usda.gov/cec/java/comb.htm
A database of all the numbers that have appeared in the National Lottery so far, and which might be good for use in the classroom, can be found at http://lottery.merseyworld.com