Part of the hook to permutations can be the impressive rate of growth of factorials. I start off with a question that steers my students toward an understanding that the size of a set dramatically increases the total number of possible permutations. The question I start with is something like the one below:
You have three marbles in a bag: 1 red, 1 blue and 1 green. How many ways can you arrange them?
If you have another bag with four marbles: 1 black, 1 yellow, 1 orange and 1 purple. How many ways can you arrange them?
If you put all the marbles together in one bag, how many ways can you arrange them?
I scaffold this question a bit with discussion. I want students to understand that they are comparing 3! + 4! to 7! The central theme is that 3! + 4! < 7!
To help students understand the amazing growth of a factorial, we refer back to a deck of cards. If we follow the logic of arrangements, students see that we can arrange a deck of cards 52! ways. This number is enormous. Plug it into any calculator and you will get around 8 x 10^67. To put this in perspective, that number is larger than the total number of known stars in the universe.
Although "larger" might not put it in the correct perspective. It is 43 powers of 10 larger. To help the kids understand this, you might say that is "a thousand million billion billion billion billion times larger."
After our really big intro to this lesson, I ask students to work through seven problems and show their work for each question. By show their work, I mean write their work using permutation notation. Since they have no background with this notation, I start with Question 7, P notation
I help students recognize that the first number denotes how many items we are arranging and the second number denotes how many "spots" we are arranging the items into. I also tell them that they can remember the "P" as meaning "permutation."
I ask them to write out, in words and as an expression, what each choice represents. For example, the answer is choice 3, which is written as 10P3 = 10 * 9 * 8, or "you have 10 items to arrange and three spots to arrange them into."
Although the other 6 questions don't ask for it, I ask them to try and write each question using similar notation. If they are unable to find a way, I ask them to skip it and review it with us during the summary at the end.
Perhaps the funnest aspect of a permutation is its connection to codes and code breaking. Question 5 taps into that context and because of that I always review it with the class.
One misconception that students face is how to set up a problem that has digits and letters. To get them started, it is important to think of the permutations of digits and then the permutations of letters and then multiply them.
With the digits, students often misinterpret 0-9 as nine available digits. We always step back and ask, "why is that wrong? how many digits are there?" Once we agree that we are dealing with 10 digits, we discuss the significance of repetition. This means that we have 10^3 arrangements, not 10*9*8.
Letters are equally confusing for me. I tell them that I feel a bit nervous whenever a question uses the alphabet, since I can never remember how many letters are in the alphabet. Fortunately they tell us that there are 26 letters in the alphabet (phew!).
Here we discuss the significance of not allowing repetition. I encourage students to write the letter permutations as 26P3 = 26*25*24
Putting this all together we have 10*10*10*26*25*24
In the second part of the question, we restrict the password by not allowing repetition of digits. This reduces the number of combinations. I ask students, before we calculate, "why does it make sense that this would reduce the number of possible combinations?"
Then we finish with the calculation:
and the difference between the two (which is easy to leave out):
10*10*10*26*25*24 - 10*9*8*26*25*24