Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 18: Probability 251

Do you see how the factorial in the denominator cancels the part of the factorial in the numerator

that you don’t want? And isn’t it good to know you don’t have to write out all those factorials all

the time?

So if you need the permutations of 7 things taken 4 at a time, the basic counting principle says

you want^7654 , and the permutation formula says 74 7

74

7

3

7654 321

321

P



!

!

!

!





.

The approaches are different, but the result is the same.

CHECK POINT

6. What is 6!?


7. What is 7!?


8. What is 7! z 6?


9. Find the permutations of 9 things taken 3 at a time.


10. Find the permutations of 10 things taken 4 at a time.

Combinations

If 8 people are candidates in a school board election and the top two finishers will serve on the

board, you might ask how many permutations of 8 people taken two at a time are possible. You

can calculate that 82 8

82

8

6

P 87 56



!

!

!

!

. There are 56 different orders of finish.

On the other hand, if both of the top two finishers will serve, regardless of which is first and

which is second, the order in which they finish doesn’t really matter. Mr. Smith and Ms. Jones is

a different permutation from Ms. Jones and Mr. Smith, but they both give you the same school

board members.

Because all the different arrangements of the same two people shouldn’t count separately, the

number of permutations is too big, but it can give you a place to start. There are 56 permutations,

or arrangements, of 8 people taken 2 at a time, but fewer combinations or groups in which order

doesn’t matter. Specifically, there are half as many combinations as permutations.

DEFINITION

A combination is a way of selecting a number of objects from a larger group when the

order of the objects is not significant.