9.1 Basic Combinatorics
Name: ______________________
Objectives: Students will be able to use the multiplication principle
of counting, permutations or combinations to count the number of
ways that a task can be done.
Isaac is a freshman at Kent State University. He is
planning his fall schedule for next year. He has the choice
of 3 math courses, 2 science courses and 2 humanities
courses. He can only select one course from each area.
How many course schedules are possible?
Apr 22­6:22 PM
Multiplication Principle of Counting
If a procedure P has a sequence of stages S1, S2, S3, ..., Sn and if
S1 can occur in r1 ways
S2 can occur in r2 ways
...
Sn can occur in rn ways,
then the number of ways that the procedure P can occur is the
product r1r2 ...rn.
So, in the case of the college course scheduling, instead of the tree
diagram, we could have:
Mar 3­1:55 PM
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Examples:
1.) In this class, how many girl-boy pairs are possible?
2.) If you toss a coin, then roll a 6-sided die and then spin a
4-colored spinner with equal sections, how many outcomes are
possible?
3.) How many Ohio license plates are possible if no letter or
number can be repeated?
Apr 22­6:45 PM
Factorial Notation: n! = n (n-1) (n-2) 3 2 1
Examples:
1.) 3! =
2.) 6! =
In the calculator: MATH - PRB - 4: !
There are n! distinguishable permutations of an n-set
containing n distinguishable objects.
Examples:
1.) How many ways are there to line up the boys in this class?
2.) There are 5 favorite runners in a race. How many ways can
the runners win 1st, 2nd, 3rd, 4th and 5th place?
3.) How many ways can the letters of the word MATH be
arranged?
Apr 22­6:59 PM
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If an n-set contains n1 objects of a first kind, n2 objects of a
second kind, and so on, with n1 + n2 + ... + nk = n, then the number of
distinguishable permutations of the n-set is:
____n!
n1! n2! ... nk!
Examples:
1.) How many ways can the letters of the word CALCULUS be
arranged?
2.) How many ways can the letters of the word MISSISSIPPI be
arranged?
Apr 22­7:28 PM
Example: There are ___ boys in this class. How many ways can
we select the positions of Math Guru, Math Genius and Math
Wizard, assuming that one person cannot hold more than one
position?
This is a permutation of ___ objects taken 3 at a time. We need
a new rule.
The number of permutations of n objects, taken r at a time, is
defined as nPr = n!
.
(n - r)!
In the calculator: MATH - PRB - 2: nPr
Note: With permutations: ORDER MATTERS
Apr 22­6:59 PM
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Example: Suppose Mrs. Meinke wants to know the number of
possible girl-girl pairs in this class. Why can't a permutation be
used?
The number of combinations of n objects taken r at a time is
defined as nCr = n!
.
(n - r)!r!
In the calculator: MATH - PRB - 3: nCr
Note: With combinations: ORDER DOES NOT MATTER.
Now, using combinations, let's find the number of possible
girl-girl pairs.
Apr 22­7:18 PM
Examples
1.) In the Miss America pageant, 51 contestants must be narrowed
down to 10 finalists who will compete on national television. In how
many possible ways can the 10 finalists be selected?
2.) Sixteen actors answer a casting call to try out for roles as
dwarfs in the production of Snow White and the Seven Dwarfs. In
how many different ways can the director cast the seven roles?
Mar 3­2:20 PM
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So far, we've been studying objects that are arranged in a line.
When objects are arranged in a circle, some of the arrangements
are alike.
Example: Consider the problem of making distinct arrangements
of 6 people sitting around a table playing cards. How many
seating arrangements are possible?
If n objects are arranged in a circle, then there are n! or (n-1)!
permutations of the n objects around the circle.
n
Apr 22­7:38 PM
Mar 8­8:13 AM
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Example: How many ways are there to place 12 decorative
symbols around the face of a clock?
Assignment: Pages 708-710: #1-41 odd, 54,55
Show all work!
Mar 3­2:27 PM
9.1 Group ICE
Name: _____________________
1.) The head of a personnel department interviews 8 people for
three identical openings. How many different groups of three can
be employed?
2.) How many 5 digit numbers exist? (We'll count numbers like
00001 as 5 digits.)
3.) How many different sequences of heads and tails are there if a
coin is tossed 10 times?
Mar 3­2:32 PM
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4.) How many ways are there to assign 10 people the roles of
president, vice president and treasurer?
5.) How many ways are there to arrange the letters in the word
STRAWBERRIES?
6.) Make up a counting problem that has 12C3 as its answer.
Mar 3­2:36 PM
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