COUNTING PRINCIPALS, PERMUTATIONS, AND COMBINATIONS
Slide 2
A COUNTING PROBLEM ASKS HOW MANY WAYS SOME EVENT CAN OCCUR. Ex.
1: How many three-letter codes are there using letters A, B, C, and
D if no letter can be repeated? One way to solve is to list all
possibilities. ABCABD ACBACDADBADC BAC BADBCABCDBDABDC
CABCADCBACBDCDACDB DABDACDBADBCDCADCB
Slide 3
Ex. 2: An experimental psychologist uses a sequence of two food
rewards in an experiment regarding animal behavior. These two
rewards are of three different varieties. How many different
sequences of rewards are there if each variety can be used only
once in each sequence? Next slide
Slide 4
Another way to solve is a factor tree where the number of end
branches is your answer. Count the number of branches to determine
the number of ways. There are six different ways to give the
rewards.
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FUNDAMENTAL COUNTING PRINCIPLE Suppose that a certain procedure
P can be broken into n successive ordered stages, S 1, S 2,... S n,
and suppose that S 1 can occur in r 1 ways. S 2 can occur in r 2
ways. S n can occur in r n ways. Then the number of ways P can
occur is
Slide 6
Ex. 2 Revisited: An experimental psychologist uses a sequence
of two food rewards in an experiment regarding animal behavior.
These two rewards are of three different varieties. How many
different sequences of rewards are there if each variety can be
used only once in each sequence? Using the fundamental counting
principle: 3 2 X 1st reward 2nd reward
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PERMUTATIONS An r-permutation of a set of n elements is an
ordered selection of r elements from the set of n elements ! means
factorial Ex. 3! = 321 0! = 1
Slide 8
Ex. 1 Revisited: How many three-letter codes are there using
letters A, B, C, and D if no letter can be repeated? Note: The
order does matter
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COMBINATIONS The number of combinations of n elements taken r
at a time is Where n & r are nonnegative integers & r <
n Order does NOT matter!
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Ex. 3: How many committees of three can be selected from four
people? Use A, B, C, and D to represent the people Note: Does the
order matter?
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Ex. 4: How many ways can the 4 call letters of a radio station
be arranged if the first letter must be W or K and no letters
repeat?
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Ex. 5: In how many ways can a class of 25 students elect a
president, vice- president, and secretary if no student can hold
more than one office?
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Ex. 6: How many five-card hands are possible from a standard
deck of cards?
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Ex. 7: In how many ways can 9 horses place 1 st, 2 nd, or 3 rd
in a race?
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Ex. 8: Suppose there are 15 girls and 18 boys in a class. In
how many ways can 2 girls and 2 boys be selected for a group
project? 15 C 2 18 C 2 = 16,065
Slide 16
ITS TIME TO COMBINE & EXPAND From a group of 7 men and 6
women, five people are to be selected to form a committee so that
at least 3 men on the committee. In how many ways can it be done?
What are the possible scenarios 3 men and 2 women, 4 men and 1
woman, or 5 men ( 7 C 3 x 6 C 2 ) + ( 7 C 4 x 6 C 1 ) + ( 7 C 5 ) -
but why do we add here? (525 + 210 + 21) 756
Slide 17
MORE EXAMPLES In how many different ways can the letters of the
word 'LEADING' be arranged in such a way that the vowels always
come together? The word 'LEADING' has 7 different letters. But,
when we pair all the vowels together (EAI) they count as 1 letter
So there are 5 letters 5! = 120 combinations But the vowels can
also be rearranged, so 3! = 6 When we combine these two we get 120
x 6 = 720 possible solutions
Slide 18
CONTINUED In how many different ways can the letters of the
word 'DETAIL' be arranged in such a way that the vowels occupy only
the odd positions? There are 6 letters in the given word, out of
which there are 3 vowels and 3 consonants. Let us mark these
positions as under: (1) (2) (3) (4) (5) (6) Now, 3 vowels can be
placed at any of the three places marked 1, 3, 5. Number of ways of
arranging the vowels = 3 P 3 = 3! = 6. Also, the 3 consonants can
be arranged at the remaining 3 positions. Number of ways of these
arrangements = 3 P 3 = 3! = 6. Total number of ways = (6 x 6) =
36.