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Permutations (Part A)
A permutation problem involves counting the number of ways to select some objects out of a group.
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There are THREE requirements
for a permutation.
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Permutation Requirements
1. The n objects are all different/distinguishable.
2. No object can be repeated.
3. Order makes a difference
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With permutations, every little detail matters.
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Permutations are for arrangements where the order of the objects matters.
Alice, Bob and Charlie is different from
Charlie, Bob and Alice.
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Telephone numbers are a good example
of a permutation.
For example, the phone number 5382783 is different from 5832784.
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A permutation is just the
fundamental counting principle
expressed as a formula.
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The number of permutations of n different objects takenr at a time is given by:
Permutation Formula
, n ≤ r
"n permutation of r"
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Note: When r = n, all of the objects in a group are selected and arranged in a specific order.
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How many different arrangements are there? List all the possible ways these three people can stand in line.
Example
Anna, Marie and Brian line up at a banking machine.
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AMB ABM
MAB MBA
BAM BMA
A list of possible arrangements are:
There are 6 possible arrangements.
A AnnaM MarieB Brian
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Sample Problems
a) 5P3
1. Evaluate.
b) 10P4
c) 8P8
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3. Suppose 1st, 2nd and 3rd place prizes are to be awarded to a group of 8 trumpeters. How many ways are there to award 3 prizes to 8 people?
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4. How many different 5 digit numbers can be made using the numbers 1, 2, 3, 4, 5, 6, 7 by only using each number once?
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5. How many ways can a president, vice president, a secretary
and treasurer be selected from a class of 25 students?
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6. How many ways can 7 books be arranged on a shelf
if they are selected from 10 different books?
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7. From 25 raffle tickets, 5 tickets are to be selected in order.
The first ticket wins $250, the second $200, the third $150,
the fourth $100 and the fifth $50. How many ways can
these prizes be awarded.
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So far, all examples of permutations have been
distinguishable permutations. This means that
all the objects are different from one another.
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Suppose you had the letters a, b, c, and d and
were asked to form all words using these four
letters. How many words can be formed?
Permutations: Part B
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Your answer would be different if the letters you had
to work with were a, a, b, c.
This would be an example of a permutation that is
nondistinguishable.
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In the 24 words formed, half will appear the same
because we cannot distinguish between the two a's.
We would only have 12 unique words.
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Permutation with Repetition
The number of permutations of n objects in which
n1 are alike, n2 alike, etc., is:
where
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1. How many different words can be formed using
the letters of the word LIBBY?
Sample Problems
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2. What is the total number of permutations of the letters
in the word BANANA?
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3. Find the number of different ways of placing 16 balls
in a row given that 4 are black, 3 are green, 7 are red,
and 2 are blue.
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4. Miss Sherrard's hockey team has 20 players consisting of
12 forwards, 6 defence, and 2 goalies. How many ways
can you arrange the players?
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5. How many different words can be formed using
the letters of the word MISSISSIPPI?
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Specific Positions
Frequently when arranging items, a particular position must be occupied by a particular item. The easiest way to approach these questions is by analyzing how many possible ways each space can be filled.
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Sample Problems
1. How many ways can Adam, Beth, Charlie and Doug be seated in a row if Charlie must be in the second chair?
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2. How many ways can you order the letters of KITCHEN if the arrangements must start with a consonant and end with a vowel?
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3. How many ways can you order the letters of UMBRELLA if the arrangements must begin with exactly two L's?
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4. How many ways can you order the letters of TORONTO if it begins with exactly two O's?
NOTE: Exactly two O's means the first 2 letters must be O, and the third must NOT be an O. Don't forget repetitions.
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Items Always Together
Sometimes, certain items must be kept together. To do these questions, you must treat the joined items as if they were only one object.
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1. How many arrangements of the word ACTIVE are there if C and E must always be together?
Sample Problems
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