The number of different
arrangements is 24 or
4! = 4 • 3 • 2 • 1. There are
24 different arrangements, or permutations, of the four students
walking side-by-side.
The notation for a permutation: n Pr n
is the total number of
objects r is the number of objects
chosen (want)
See how to use
your
TI-83+/TI-84+ graphing calculator with
permutations. Click calculator.
The formula for a
permutation:
OR
The formula
for a permutation:
(Remember that 0! = 1.)
(Note: if
n = r, as it did with the students walking
side-by-side, then n Pr =
n! for either formula.)
Examples:
1.
Compute: 5 P 5
5 · 4 · 3 · 2 · 1 = 120
2.
Compute: 6 P 2
6 · 5
= 30 or
multiply
by two factors
of the factorial,
starting with 6
3. Find
the number of ways to arrange 5 objects that are chosen from a set
of 7 different objects.
7 P 5
= 7·6·5·4·3
= 2520 or
4.
What is the
total number of possible 5-letter arrangements of the letters
w, h, i, t, e, if each
letter is used only once in each arrangement?
5 P5 =
5·4·3·2·1
= 120 or
or simply 5!
5.
How
many different 3-digit numerals can be made from the digits
4, 5, 6, 7, 8 if a digit can appear
just once in a numeral? 5 P3
= 5·4·3
= 60 or
Permutations with Special Arrangements:
Example:
Using the letters in the word "
square
", tell how many 6-letter arrangements, with no repetitions,
are possible if the :
a)
first letter is a vowel.
b)
vowels and consonants alternate, beginning with a consonant.
Solution: Part a: Hint:
When working with
"arrangements", it is often helpful to put lines down to represent
the locations of the items.
For this problem, six "locations" are
needed for 6-letter arrangements.
_____ • _____ •
_____ • _____ • _____ •
_____
The first locations must be a
vowel (u, a, e). There are three ways to
fill the first location.
__3___
• _____ • _____ •
_____ • _____ • _____
After the vowel has been placed in the first
location, there are 5 letters
left to be arranged in the remaining five spaces.
__3__
•
__5__ •
__4__ •
__3__ •
__2__ •
__1__
or
3 · 5P5 =
3 · 120 = 360
Part b:
Six locations are needed for the
6-letter arrangements.
_____ • _____ •
_____ • _____ • _____ • _____
Beginning with a consonant, every other
location must be filled
with
a consonant. (s, q, r
)
__3__
• _____ •
__2___ •
_____ • ___1__
• _____
The remaining locations are filled with
the remaining three
vowels:
__3__ •
__3___ • __2___
• __2___ •
___1__ •___1__
= 36
multiply
by two factors
or simply 5!
