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Math 166, Fall 2015, Robert
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2.1 and 2.2- Permutations and Combinations
The Multiplication Prinicple: Suppose you are given a two tasks to do. In
your first task, you must decide between n1 possible choices. In the second task,
you must decide between n2 possible choices. Then there are a total of
possible ways for you to complete the tasks.
Note: the multiplication prinicple can be extended to any number of tasks. If
you have k tasks, the first of which consists of n1 choices, the second consists of
n2 choices, and so on; then there are a total of n1 · n2 · · · · · nk possible ways in
which the tasks can be completed.
Example 1 The manager of a fast food chain wants to brag about how many
combo meals a customer can make from the items on their menu. The chain
offers 12 different drinks, 10 different entrees, and 6 different sides. If a customer’s are allowed to make combo meals by combining exactly one item of each
type, how many combo meals can be made?
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Math 166, Fall 2015, Robert
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Example 2 Using the English alpahabet and without regard to spelling, how
many three letter words can be made
• without restrictions?
• if the first letter must be a vowel?
• if the same letter cannot be used twice in a row?
Example 3 Let A be a set with n elements. How many subsets does A have?
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Math 166, Fall 2015, Robert
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Factorial: For any natural number n, we define n! as follows:
• 0! = 1
• If n > 0, then n! = n · (n − 1) · (n − 2) · · · 3 · 2 · 1
A permutation of a set of elements is an ordered arrangement of all of the
elements. The number of permutations of n distinct objects is
P (n, n) =
The number of permutations of r objects taken from a set of n objects is
P (n, r) =
Example 4 An ice hockey team consists of 6 players. If there is only enough
room for one of them to skate onto the ice at a time, how many ways can the
players skate onto the ice?
Example 5 A group of 12 politicians form a committee. The committee must
have a president, a vice-president, a secretary, and a treasurer. In how many
ways can these positions be filled?
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Math 166, Fall 2015, Robert
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Example 6 Spencer and Tyler decide to go watch a movie with three friends.
How many ways can these 5 friends sit if
• there are no restrictions?
• Spencer and Tyler sit beside each other?
• Spencer and Tyler do not sit beside each other?
Example 7 A student decides to get a bookshelf to organize his school books.
He has 5 math books, 4 history books, and 3 economics books.
• How many ways can the student organize these books onto a single shelf ?
• How many ways can the student organize these books onto a single shelf if
he decides to keep all of the books in the same subject together?
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Math 166, Fall 2015, Robert
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Example 8 A state uses license plates with 2 numbers followed by 4 letters.
The state does not allow license plates to have repeated letters or digits. How
many plates can be made?
A combination of r distinct objects taken from a set of size n is a selection of
r objects without regard to order. The number of such combinations that can
be formed is
C(n, r) =
Example 9 A dozen doughnuts are placed in a row from a large collection of
glazed and chocolate doughnuts. In how many ways can the doughnuts be placed
if 7 chocolate doughnuts are selected?
Example 10 A total of 15 people are called for jury duty, 7 male and 8 female,
from which a 12-person jury will be formed.
• How many juries can be formed?
• How many juries can be formed if it is required that there be an equal
number of men and women on the jury?
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Math 166, Fall 2015, Robert
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Example 11 In the game poker, players make hands out of 5 cards from a
standard deck of 52 playing cards.
• How many different hands can be dealt?
• How many hands can be dealt such that the player has exactly two queens
and two kings?
• How many hands can be dealt such that the player has exactly three aces?
Example 12 Thirty students enter a local science fair. The science fair awards
one student first place, one student second place, one student third place, and
four students honorable mention. How many ways can the fair hand out these
awards?
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