MATH 141-501
Section 6.4 Lecture Notes
Permutations are arrangements of a set of objects where order matters.
Combinations are arrangements of a set of objects where order doesn’t matter.
Permutations
A permutation is an arrangement of objects in a definite order.
List all permutations of the set A = {1, 2, 3}. How many are there?
How many permutations of the set B = {1, 2, . . . , 10} are there? (Hint:
Remember the multiplication principle.)
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Number of Permutations
The number of permutations of a set with n elements is n! = n(n − 1)(n −
2) . . . 2(1).
n! = “n factorial” = “all the numbers from n counting down to 1, multiplied
together”.
Example: 10! =
(this is the number of permutations of a set with n objects)
How to calculate n! : n [MATH], [PRB], [4], [ENTER]
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Permutations of r objects from a collection of n DISTINCT objects
Idea: First choose r objects from a set of n objects, then count the number
of ways that those r objects can be ordered.
Mathematical Formula:
P (n, r) =
n!
(n − r)!
In the Calculator: n [MATH] PRB, 2: nPr r [ENTER]
Example: How many 4 digit numbers can be formed from the digits 0-9 if
no repetitions are allowed?
Example: How many different ways can a President, Vice-President, Secretary, and Treasurer be chosen from a group of 50 students?
Example: In a running event at the Olympics, there are 15 runners. How
many different possibilities are there for the medals to be awarded?
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Permutations of n objects NOT ALL DISTINCT
If the objects being arranged are not all distinct, then a different formula is
used.
Suppose we have n objects. Suppose n1 objects are alike, n2 objects are
alike, . . ., nk objects are alike, where n1 + n2 + . . . + nk = n. Then the number
of permutations of those n objects is
n!
(n1 !)(n2 !) . . . (nk !)
Example: How many “distinguishable ways” can the letters in the word
BOOKKEEPER be arranged?
Example: How many different seven-digit phone numbers can be made
from the digits 1,1,1,3,3,5,5?
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Combinations
The number of ways we can select r objects from a set of n objects when
order does not matter is given by
C(n, r) =
n!
r!(n − r)!
This is the number of combinations of a set of n objects, taken r at a time. It
is often called “n choose r”.
Note that we need r ≤ n for this to make sense.
CALCULATOR: n [MATH] PRB 3:nCr r [ENTER]
Example: Suppose that a presidential candidate chooses 4 states at random
in which to give speeches. How many different possibilities are possible?
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Combinations – examples
Example: In a club with 15 freshmen and 13 sophomores, how many different 4 person committees can be made with...
... 3 freshmen and 1 sophomore?
...exactly two freshmen?
...at least two freshmen?
Example: Out of a group of 100 people, 7 are left-handed. How many
samples of 10 people have...
... exactly 1 left-handed person?
...at least 1 left-handed person?
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