Welcome to this short ‘insights video’ where we are going to look at arrangements, permutations and combinations and some of the challenges learners face in solving these kind of problems.
A common misconception when sorting, or arranging objects, is to think:
‘Aha - there are six objects, so I could start sorting by choosing any one of the six. Then, there are five left, so I could choose any one of the five and so on…
...giving 6 choices plus 5 choices plus 4 choices plus 3 choices plus 2 choices plus 1 choice…
...So that makes 21 ways…
...Which is incorrect. This error of ‘adding’ instead of ‘multiplying’ means they have not really grasped the mathematical process of making multiple selections.
Using simple examples of ‘selections’ quickly shows learners how to build up a general mathematical rule to the problem of arrangements, and then applying this rule is so much quicker, than listing all the possible outcomes particularly for more complex problems
Then learners are not always clear about the difference between a question asking then to make a selection, and making a selection in a particular order.
For example, a question about competitors in a schools sports competition. Guessing who will win the first three places is hard, but guessing the winners and the order they will win in is harder still The chance or ‘probability’ of guessing the winners in the order right too is less than just guessing the winners.
Examples like this let learners see that choosing, or ‘selecting’, from a series of options, is a very different answer from choosing, or ‘selecting’, from a series of options in a particular order!
They need to decide: are they being asked ‘how many ways they can select particular objects (using combinations) or how many ways they can arrange particular objects (and use permutations).
Finally, they must answer using the correct notation and correct formula when solving problems like this.
Learners often use nCr when they mean nPr, from not understanding the topic completely.
I hope this short insights video on permutations and combinations has been useful to you and your learners.