Mathematics Ext 1
Understanding Permutations and
Combinations I
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•
Permutations
•
Permutations with repetitions
Mathematics Extension 1: Permutations and Combinations I
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Mathematics Extension 1: Permutations and Combinations I
Permutations
A permutation is an arrangement of objects from a certain set. For example, ABC, ADB and
BAC are permutations of 3 letters taken from the letters ABCDE.
Two principles underlie even the most sophisticated counting methods.
Suppose that a choice can be made from
options and another choice can be made from
options. Then
•
The number of ways to make either the first choice or the second choice is
+ ,
assuming that all the options are different (the addition principle)
•
The number of ways of making the first choice and the second choice is
× ,
assuming that all combinations of the two choices are allowed (the multiplication
principle)
Examples
1. A menu lists 7 different main courses and 5 different desserts.
(a) If I order either a main course or a dessert, how many options do I have?
(b) If I order a main course and a dessert, how many options do I have?
2. How many ways are there to travel from A to C?
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Mathematics Extension 1: Permutations and Combinations I
When there are restrictions, deal with them first and use common sense to break the
problem down into simpler cases.
3. We wish to construct 3 letter “words” from the English alphabet. How many choices are
there, if
(a) Any choice of letters is permitted?
(b) The word must contain one vowel?
(c) The word must contain at least one vowel?
(d) The word must come before EGG in alphabetical order?
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Mathematics Extension 1: Permutations and Combinations I
Problems
1. How many ways can you arrange the letters ABCD?
2. How many three letter words can you create from the letters ABCDEF?
3. Without simplifying your answer, how many three digit numbers can be created with:
(a) No restrictions?
(b) No repetitions?
(c) Only odd numbers?
(d) Only prime numbers and no repetitions?
(e) First digit is one and no repetitions?
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Mathematics Extension 1: Permutations and Combinations I
4. How many ways can you colour the following flag if you have red, blue, yellow and green,
given:
(a) No restrictions?
(b) That you can’t use the same colour twice?
(c) That you can’t have the same colour next to each other?
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Mathematics Extension 1: Permutations and Combinations I
The number of permutations of
objects selected from
=
−1
−2 …
−
different objects is presented by:
−1
We can use factorials to simplify our notation, where
!=
Hence, we can rewrite
−1
−2 …3 ×2 × 1
in terms of factorials:
=
!
− !
Examples
1. How many three letter ‘words’ can be created from the alphabet? Leave answer in factorial
form.
2. A band of pirates has 13 members, including the captain’s parrot. A crew photo is to be
taken with seven pirates standing in a row and six sitting on chairs in front of them.
Suppose that we don’t care who stands where, but we do care who sits on which chair. In
how many possible ways can pirates be seated on the chairs? What if the captain and the
parrot must sit on the middle two chairs?
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Mathematics Extension 1: Permutations and Combinations I
Permutations with repetitions
Suppose that there are
objects, and there is ℓ of the first type of objects, ℓ of the second
type, …, and ℓ of the final type. In other words, there are
distinct types of objects. Then
the number of permutations of those objects is:
!
ℓ !× ℓ !× …× ℓ !
Examples
1. Write out all the ways of arranging the letters of LASS. Why are there not 4! permutations?
2. How many permutations of the following words?
(a) AZAM
(b) OOFT
(c) LITTLE
(d) PARRAMATTA
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Mathematics Extension 1: Permutations and Combinations I
Techniques for counting permutations
Taking cases
Sometimes a problem can be broken down into simpler cases, and you just count the
permutations of each case, and add them together.
Objects need to be together
When a question specifies that certain objects need to be together, then:
1. Group up the objects that need to be together into a block.
2. Find the permutations of the objects, treating the block as one object.
3. Find the permutations of the objects within the block.
Certain objects need to be in a certain order but not necessarily together
When this happens, take a fraction of the total amount that has the specified order.
Objects need to be separated
In the case where two objects need to be separated, find the number of ways the two
objects are together, and subtract it from the total.
In the case where more than two objects need to be separated, use the “gap method”.
1. Arrange everything but the objects that need to be separated first.
2. Find the number of “gaps” that can be chosen to place the unplaced objects.
3. Place the objects into these gaps, ensure you take repetition into account.
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Mathematics Extension 1: Permutations and Combinations I
Examples
1. Given 3 girls, Anna, Beryl and Carol, and 3 boys, Doug, Edward and Fred, how many ways
are there to arrange them in a line if:
(a) There are no restrictions?
(b) Anna must be first in line?
(c) Anna must be first in line and Fred must be last in line?
(d) There must be a boy at each end?
(e) Boys and girls must alternate?
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(f) Anna and Beryl must be together?
(g) The girls must be together and the boys must be together?
(h) The girls must be together?
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(i) Anna must be before Beryl (not necessarily next to each other)
(j) Anna must be before Beryl who must be before Carol (not necessarily next to each
other)
(k) Anna and Doug must not be together?
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Mathematics Extension 1: Permutations and Combinations I
(l) Anna, Doug and Carol cannot be together?
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Mathematics Extension 1: Permutations and Combinations I
Problems
1. How many arrangements of the word LITTLE are there if:
(a) There are no restrictions?
(b) The Ls have to be together?
(c) The two Ls are together and two Ts have to be together?
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Mathematics Extension 1: Permutations and Combinations I
(d) The Ls have to be separated?
(e) The Ts occupy the ends?
(f) An L occupies the first position?
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Mathematics Extension 1: Permutations and Combinations I
(g) I is to the left of E?
(h) The vowels are not together?
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Mathematics Extension 1: Permutations and Combinations I
2. How many 5 letter words can be formed by using the letters of the word LASSES?
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Mathematics Extension 1: Permutations and Combinations I
3. How many arrangements of the word REUBEN are there if:
(a) The Es are together?
(b) The Es are separated by 1 letter?
(c) The Es are separated by 2 letters?
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Mathematics Extension 1: Permutations and Combinations I
(d) The Es are separated by 3 letters?
(e) The R is somewhere between the two Es?
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Mathematics Extension 1: Permutations and Combinations I
4. Challenge. How many ways are there of getting from the top left hand corner to the
bottom right hand corner if you can only move downwards and rightwards?
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