Notes on Counting of Multiple Events (1) Name______
Standards Addressed: MA1D1: Students will determine the number of outcomes related to a given event.
a) Apply the addition and multiplication principles of counting.
b) Calculate and use simple permutations and combinations.
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1. A tennis team consists of three boys (John, Steve, and Ron) and two girls (Molly and Tina). In the final match of a tournament, suppose the coach of the team must select one player from his team to play. How many choices does the coach have?
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2. The same tennis team plays in a different tournament. The coach must now put one boy and one girl together on the court to play a mixed doubles match. How many unique pairings could the coach make to put on the court?
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The most important difference to notice between the problems from the previous days and the problems today is the following: the previous days' problems involved only
______event/group, but today's problems involve ______events/groups.
In both of the problems above, there are two groups of items (boys and girls). In previous problems, boys and girls would have been lumped together into one group (ex. five racers).
What is the difference between how the coach is selecting players in the two problems above?
What mathematical operation is performed with the numbers of boys and girls in order to obtain the answer in the both problems?
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The addition principle of counting states that if one event can occur m ways, and another event can occur n ways, then the number of different ways in which one of the events can occur is equal to .
The key ideas that usually suggest a problem requires the addition principle of counting are the use of the word OR and selecting ONE TOTAL ITEM. In Question 1 above, do you see either of the key phrases?
In Question 1, there are two events. The coach can select one boy, OR the coach can select
one girl. Each of these choices is treated as a separate event (problem), and then the results are added at the end because the coach can only do one of them.
A)How many ways can one boy be selected? Write using combination shorthand.______
B)How many ways can one girl be selected? Write using combination shorthand.______
Now, add A) and B) to find the total number of options for the coach.______
The multiplication principle of counting states that if one event can occur p ways, and another event can occur q ways, then the number of different ways in which both of the events can occur is equal to . This principle is also known as the fundamental counting principle.
The key ideas that occasionally suggests a problem requires the multiplication principle of counting are the word AND and selecting ONE ITEM FROM EACH GROUP. What information from Question 2 hints that it will require the multiplication principle of counting?
In Question 2, there are two events. The coach selects one boy, AND the coach selects
one girl. Each of these choices is treated as a separate event (problem), and then the results are multiplied at the end because the coach must do both of them.
A)How many ways can one boy be selected? Write using combination shorthand.______
B)How many ways can one girl be selected? Write using combination shorthand.______
Now, multiply A) and B) to find the total number of options for the coach.______
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3. Suppose someone wants to pick one favorite sports' team to watch during the year. If there are 30 NBA teams, 30 MLB teams, and 32 NFL teams, and the person can only select one team total, then how many different options are there?
4. A couple that has been married for ten years decides that they are wanting to start
a diversified portfolio of stocks. If they are looking at eight technology stocks,
seven pharmaceutical stocks, five financial stocks, and ten energy stocks, and they
wish to select one from each of these groups, how many different portfolios are possible?
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5. A tennis team consists of three boys (John, Steve, and Ron) and two girls (Molly and Tina). Suppose that the coach can only take two boys and one girl to a tournament
in Orlando. How many choices does he have?
In this question, there are still ______events. The coach must select two boys and select one girl. How many of these events will he do? Therefore, this problem involves the ______principle of counting.
A)How many ways can two boys be selected? Write using combination shorthand.______
B)How many ways can one girl be selected? Write using combination shorthand.______
Now, multiply A) and B) to find the total number of options for the coach.______
Notes on Counting of Multiple Events (2) Name______
6. Alex has just finished using all of his tokens at Adventure Crossing, and he now will exchange his tickets for lame prizes. After adding up all of his tickets and considering all of the options, he decides he can get four different-flavored Jolly Ranchers or two different-flavored Dum Dums. If there are twelve flavors of Jolly Ranchers and eleven flavors of Dum Dums, then how many choices does he have for his prizes?
7. A couple goes to TGI Friday's on a date. There are eight appetizers, 24 main courses, and six desserts from which to choose. If the couple will share two appetizers, two main courses, and one dessert, then how many unique meals could they share?
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Homework on Counting of Multiple Events
1. A gym student is forced to pick four other students to play basketball with him on his
team. If he has eleven students from which to pick, then how many different teams
can be formed?
2. At Lakeside High School, four teachers teach Math I, five teachers teach 9th grade
English, and three different teachers teach Biology. If a freshman student wanted all
three of these classes on his/her schedule, then how many unique groupings of teachers
are possible?
3. An honor council is being formed at Westmore High School that will consist of three
Juniors and five Seniors. If there are currently 15 Juniors and 20 Seniors being
considered for the positions, how many different honor councils could be formed?
4. There are eleven teams in the Big Ten in college football (I know, it doesn't make sense).
In how many different ways can the eleven teams finish from first to eleventh place if
there can be no ties?
5. A group of three friends come together to watch a movie on Friday night, but they each
have a different set of movies they want to watch. Suppose that Crystal wants to see one
of the ten movies at Regal Cinemas. Next, suppose that Jeremy wishes to spend less
money, so he recommends one of the eight movies at Masters Value Cinemas. Lastly,
Tyrone just wants to stay home and watch one of the 211 movies that are On Demand. If
the friends are only watching one movie, how many options do they have?
6. Zed goes to Wendy's with six dollar bills. He is able to afford five items off of the
99 cent value menu, or he could choose to simply order a combo meal. If there are
nine items on the value menu and twelve different combo meals, then how many
options does he have?
7. At a computer store, suppose there are six Dell computers, four Apple computers,
and seven HP computers. If Yolanda is only buying one computer at this store, then
how many choices does she have?
8. At Wife Saver, a child's plate consists of one meat, two sides, one bread, and one drink.
If there are five choices of meat, sixteen choices of sides, four choices of bread, and nine
choices of drink, how many different child's plates could be formed?
9. Jack wants to fly from Augusta, GA to the state of Washington, and he is hoping to have
exactly three stops on his journey. Suppose the flight leaving Augusta can fly to either Atlanta or Charlotte. Next, each of these flights have five different destinations. Finally, flights leaving this last set of destinations fly to two places in Washington: Seattle and Spokane. How many different trips could be formed from Augusta to Washington?
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10. A)Three people enter a race. How many different ways can the racers finish from
1st place to 3rd place?
B)Four people are standing in line at a water fountain. How many unique lines could
be formed?
C)How many different ways can five people stand in a line?
D)Other than these problems involving permutations, what other patterns do you see?
Think specifically about how the problem is solved.
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Answers:
1. 330 2. 60 3. 7,054,320 4. 39,916,800 5. 229 6. 138
7. 17 8. 21,600 9. 20 10. a) 6 b) 24 c) 120 d) In n = r, then the answer is n