Idiot\'s Guides Basic Math and Pre-Algebra

250 Part 4: The State of the World

This question of the number of ways you can arrange a group of things—numbers, letters,

people, whatever—is the question of how many permutations there are. The word permutation

means arrangement or ordering.

DEFINITION

A permutation is an arrangement or ordering of a group of objects.

You can answer the question about the number of permutations by using the basic counting prin-

ciple. If you have to put five digits in order, there are 5 v 4 v 3 v 2 v 1 = 120 permutations.

This pattern of multiplication, from a number (in this case, 5) down to 1, is called a factorial.

The symbol for factorial is an exclamation point. 5! = 5 v 4 v 3 v 2 v 1 = 120.

The number of orders in which the 8 children on a soccer team could line up at the water

fountain, or the number of different arrangements of 8 people taken 8 at a time, is 8! =

8 v 7 v 6 v 5 v 4 v 3 v 2 v 1 = 40,320. For any number n, the permutations of n things taken

n at a time is n!

But what if you’re not taking n things at a time? For example, a baseball team needs to put 9

players on the field but might have 25 players on the roster. Now, if you know anything about

baseball, you know that the players aren’t all interchangeable, but still, how many permutations

of 25 people taken 9 at a time are possible?

Your best strategy is to go back to the basic counting principle. Make nine slots and start filling

them from 25 on down. 25 v 24 v 23 v 22 v 21 v 20 v 19 v 18 v 17. Multiply that out and

you get } 7.4 v 1011. That’s approximately 740 billion possible lineups. Maybe it’s a good thing

the players aren’t all interchangeable.

You’re still talking about an arrangement, or permutation, but an arrangement of 9 out of the 25,

not all 25. You don’t want the entire 25!, but you do want the first 9 slots of it. You can write a

formula for the permutations of n things taken t at a time as

ntP

n

nt





!

!

In this case, the permutations of 25 things taken 9 at a time is

25 9

25

25 9

25

16

25 24 23 22 21 20 19 18 17 16 15 14

P



!

!

!

!

13 12 11 10 9 8 7 6 5 4 3 2 1

16 15 14 13 12 11 10 9 8 7 6654321