Math 1313
Section 6.3
Section 6.3 Generalized Multiplication Principle
Suppose a task T1 can be performed in N1 ways, a task T2 can be performed in N2 ways,…, and ,
finally a task Tn can be performed in Nn ways. Then the number of ways of performing the tasks T1,
T2, …, Tn in succession is given by the product
N1• N2•… •Nn
Example 1: A coin is tossed 3 times, and the sequence of heads and tails is recorded.
a. Determine the number of outcomes of this activity.
b. List the outcomes of this experiment by first drawing a tree diagram.
Example 2: The Burger Bar offers the following items on its menu:
Burger
Single Meat
Double Meat
Sides
Fries
Onion Rings
Fruit Bowl
Cheddar Peppers
Beverages
Tea
Coffee
Soda
Desserts
Cheesecake
Brownie
Cookie
Ice Cream Cone
If a customer chooses 1 item from each category, how many meals can be made? List 1 meal
possible.
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Math 1313
Section 6.3
Example 3: An identification number for employees at a certain company contains six digits. How
many ID numbers are possible if repetition is allowed?
Example 4: A license plate consists of 2 letters followed by 4 digits. How many license plates are
possible if the 1st letter can't be O, the 1st digit can't be 0 and no repetitions are allowed?
Example 5: In the original plan for area codes in 1945, the first digit could be any number from 2
through 9, the second digit was either 0 or 1, and the third digit could be any number except 0.
With this plan, how many different area codes were possible?
Example 6: Six performers are to present their comedy acts on a weekend evening at a comedy
club. One of the performers insists on being the last stand-up comic of the evening. If this
performer’s request is granted, how many different ways are there to schedule the appearances?
Example 7: The call letters for radio station begin with K or W, followed by 3 additional letters.
How many sets of call letters having 4 letters are possible? Repetition is allowed.
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Math 1313
Section 6.4
Section 6.4: Permutations and Combinations
Definition: n-Factorial
For any natural number n, ݊ሺ݊ − 1ሻሺ݊ − 2ሻ … 3 ∙ 2 ∙ 1
0! = 1
A permutation is an arrangement of a specific set where the order in which the objects are
arranged is important.
௡!
Formula: ܲ ሺ݊, ‫ݎ‬ሻ = ሺ௡ି௥ሻ! , ‫݊ ≤ ݎ‬
where n is the number of distinct objects and r is the number of distinct objects taken r at a time.
Formula: Permutations of n objects, not all distinct
Given a set of n objects in which ݊ଵ objects are alike and of one kind, ݊ଶ objects are alike and of
another kind,…, and, finally, ݊௥ objects are alike and of yet another kind so that
݊ଵ + ݊ଶ + ⋯ + ݊௥ = ݊
then the number of permutations of these n objects taken n at a time is given by
݊!
݊ଵ ! ݊ଶ ! … ݊௥ !
A combination is an arrangement of a specific set where the order in which the objects are
arranged is not important.
Formula: ‫ ܥ‬ሺ݊, ‫ݎ‬ሻ =
௡!
௥!ሺ௡ି௥ሻ!
, ‫݊≤ݎ‬
where n is the number of distinct objects and r is the number of distinct objects taken r at a time.
Example 1: You are in charge of seating 5 honored guests at the head table of a conference. How
many seating arrangements are possible if the 5 chairs are on one side of the head table?
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Math 1313
Section 6.4
Example 2: Find the number of ways 9 people can arrange themselves in a line for a group
picture.
Example 3: In how many ways can 7 cards be drawn from a well-shuffled deck of 52 playing
cards?
Example 4: An organization has 30 members. In how many ways can the positions of president,
vice-president, secretary, treasurer, and historian be filled if not one person can fill more than one
position?
Example 5: An organizations needs to make up a social committee. If the organization has 25
members, in how many ways can a 10 person committee be made?
Example 6: If there are 40 contestants in a beauty pageant, in how many ways can the judges
award 1st prize and 2nd prize if not one person can be awarded 1st and 2nd?
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Math 1313
Section 6.4
Example 7: In a production of West Side Story, eight actors are considered for the male roles of
Tony, Riff, and Bernardo. In how many ways can the director cast the male roles?
Example 8: How many permutations can be formed from all the letters in the word MISSISSIPPI.
Example 9: A museum of fine arts owns 8 paintings by a given artist. Another fine arts museum
wishes to borrow 3 of these paintings for a special show. How many ways can 3 paintings be
selected for shipment out of the 8 available?
Example 10: A certain company has to transfer 4 of its 10 junior executives to a new location,
how many ways can the 4 executives be chosen?
Example 11: A student belongs to a entertainment club. This month he must purchase 2 DVDs
and 3 CDs. If there are 10 DVDs and 10 CDs to choose from, in how many ways can he choose his
5 purchases?
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