Calculating Permutations

You've just transferred to a new (and rather strange) middle school, which offers the following 10 classes:

Armadillo Riding 101 (A)
Banana Topology 101 (B)
Car Theft 101 (C)
Doll Beheading 101 (D)
Egg Throwing 101 (E)
Frog Dissection 101 (F)
Grape Swallowing 101 (G)
Hamburger Flipping 101 (H)
Igloo Building 101 (I)
Jello Nailing 101 (J)

If you will have 6 of these classes on your schedule, how many different schedules are possible?

First, note that the order of the classes matters. The schedule ABCDEF is different from FEDCBA. You don't just care which classes you take; you also care which is first hour, which is second hour, etc. Because the order matters, mathematicians would say that you want to find the number of 6-class permutations. (If you just wanted to choose 6 classes regardless of their order on your schedule, you would calculate combinations instead.)

To calculate the number of permutations in this problem, draw 6 blanks on your scratch paper, each representing an hour of the day. They are blanks because each could represent A, B, C, D, E, F, G, H, I, or J.
_ _ _ _ _ _

Start with the first blank, which stands for first hour. How many possible choices do you have to fill that blank? 10, because you can choose from 10 classes. So write 10 in that blank.
10 _ _ _ _ _

Now for the second blank. How many choices do you have to fill that blank? Only 9, because you can choose any of the 10 classes EXCEPT for the one that you already picked (for first hour). So far you have 10*9=90 possible schedules because for each of the 10 choices for first hour, there are 9 choices for second hour.
10*9 _ _ _ _

Now for the third blank. Do you see the pattern yet? You have 8 choices for this blank because you can choose any of the 10 classes EXCEPT for the two that you already picked (for first and second hours). Your total so far is 10*9*8=720 possible schedues.
10*9*8 _ _ _

Repeat the process for the last three blanks. Eventually, you will end up with the expression
10*9*8*7*6*5
= 151,200 schedules!

Let's generalize now. To find the number of permutations of n (10 in our example) things chosen k (6 in our example) at a time,

Draw (or picture mentally) k blanks.
Fill the first blank with n.
Fill the second blank with n-1, the third with n-2, and so on until you reach the last blank.
Multiply the numbers in all the blanks to obtain the total number of permutations.

In other words, the number of permutations of n things chosen k at a time is

n(n-1)(n-2)...(n-k+1)

which can be written in factorial form as