Probability And Statistics Worksheets Page 24
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42An arrangement of items in a particular order is called a permutation. For the four
letters M, A, T, H, there are 24 different 4-letter permutations.
MATH
MAHT
MHAT
MHTA
MTHA
MTAH
AMTH
AMHT
AHTM
AHMT
ATHM
ATMH
TAMH
TAHM
TMAH
TMHA
THAM
THMA
HAMT
HATM
HMAT
HMTA
HTAM
HTMA
You can use the fundamental counting principle to find the number of
permutations of any group of items.
For each arrangement of M, A, T, H, there are 4 choices for the first letter, 3
choices for the second, 2 choices for the third, and 1 choice for the fourth.
4 3 2 1
24.
4 3 2 1 can be written in factorial notation as 4!
In general, the number of permutations of n different items is written ! and read
as factorial.
E x a m p l e 2
In how many different ways can you arrange your math,
science, social studies, language arts, and literature anthology
books in a row on a shelf?
Solution
There are five books. Find the number of permutations of five items.
number of permutations of five items
5!
5 4 3 2 1 or 120
There are 120 different ways to line up five books on a shelf.
Sometimes you need only part of a set of items, such as selecting
two of nine players. The number of permutations of n different
Check
items, taken r items at a time, with no repetitions, is written
P
.
Understanding
n
r
Use the formula below to find the number of permutations when
only part of a set is used.
What is wrong with the
notation
P
?
4
5
n !
P
n
r
(n
r)!
E x a m p l e 3
SPORTS
Eight teams enter a tournament. How many different arrangements of
first-, second-, and third-place winners are possible?
Solution
n !
Use the formula:
P
n
r
(n
r)!
8!
P
8
3
(8
3)!
8 7 6 5 4 3 2 1
Cancel common factors to
8 7 6
336
simplify the computation.
5 4 3 2 1
There are 336 ways for teams to finish first, second, and third.
403
Lesson 9-5 Permutations and Combinations
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