Permutations

An arrangement of all or part of a number of things in a definite order.

Order of the Arrangement Counts!!!

Permutations are used to generate:- license plate numbers- phone numbers- social insurance numbers- locker combinations

We show permutations with the symbol "nPr" which reads

the number of permutations of "n" things arranged or taken or picked "r" at a time

For example, How many 2-letter "words" can be made using the letters of the alphabet?

Two solutions possible

Ex)(a) How many permutations can be formed from the letters in the word JUSTICE using all 7 letters?

In general, nPr = n! n r (n - r)! n N

Using the above example..

(b) How many permutations can be formed from JUSTICE using only 5 letters at a time?

or 7P5 says "using 7 objects picking or arranging them 5 at a time"

Also reads "7 pick 5"

On calculator 7P5 =

Or 7P5 =

Ex)A group of 9 different books are to be selected and arranged on a shelf for display. How many arrangements are possible?

or

9P4 =

Note: Arrangement, lineup, selected suggest order or perms.

Ex) Solve for n, given nP2 = 30

Using nPr to solve for n or r

Ex) Solve for r, given 5Pr = 20

Perms with Repeating Objects

Ex) How many ways can you arrange the letters of "WOW"?

We can only tell 3 apart.The other 3 are not distinguishable.

Since the W repeats twice, we correct by dividing by 2!

Perms with RestrictionsNote: Must ALWAYS fill restrictions first!!

Ex) Using the first 10 letters of the alphabet, how many different arrangements can you make if you must

(a) start with a vowel

(b) start and end with a vowel

(c) Not start or end with a vowel

Ex) Using all the letters of the word BRAINS (no reps), how many 4-letter arrangements are possible

(a) ending in "R"

(b) with consonants only

(c) with consonants and vowels alternating

Perms with Cases

Ex) If there are six chairs in a row, how many ways can 3 boys and 3 girls sit if they must alternate?

Ex) How many different four-digit numbers can be made that are even using the digits 0, 1, 2, 3, 4, 5, 6 ?

Ex) How many different 3-digit numbers can be made using the digits 2, 3, 4, 5, and 0 if the number must be divisible by 5?

Ex) How many different numbers of at least three digits can be formed from the integers 1, 2, 3, 4, 5?

Ex) How many ways can you arrange the letters of WOWW?

In general, the number of perms of n objects with r identical objects is n!

r!

Ex) Using all letters in the AARDVARK how many different arrangements are possible?