Intro: Show the following video:
"How would we determine the total number of ways the book shelf could be arranged?"
"What would we need to know?"
After some discussion, show the picture of the top left shelf as seen below:
Then I would ask the following questions:
1) How many different ways can these books be arranged, if the spine of the book must be facing out, and:
a) The shelf will only have 4 books and a pink book must be on each end?
b) The shelf will only have 8 books the pink books must be next to each other?
c) All books are on the shelf with no restrictions?
The last question is there to assist with introducing factorial notation.
Explain that: "!" is the symbol for factorial notation and can be used on any non-negative integer n. The formal defintion is:
As well as how to get use it on the calculator.
I would then go through operations on factorials showing that (n!)(x!) does not equal (nx)! nor does it hold for any other operation. As well that 0! = 1
Next ask for the previous picture:
2) How many different ways can you arrange the books on the shelf if you only want:
a) 4 books on the shelf?
b) 7 books on the shelf?
c) 13 books on the shelf?
Using the process, you can introduce nPr, and how the previous answers can be solved with n!/(n-r)!, with guiding the students towards this answer.
The formal definition being:
When permutating n objects picking r at a time we would write nPr = n!/(n-r)!
Ask the students how many diffeerent ways to arrange the letter FILE, then FILL, then FLLL, then LLLL.
Let students work and struggle through, and lead them towards the identity of:
Organizing a objects where there are n repetitions of one object, there are a!/n! distinct permutations.
From here I have yet to find real life scenarios and would give various questions and words to rearrange such as MISSISSIPPI. (Would love ideas here if you have any...)