Super Subs makes three types of sandwiches: chicken salad, turkey or grilled cheese. Costumers then choose a side: chips, French Fries or fruit cup. And for a drink, they can have soda or water.
The tree diagram below shows all possible combinations.
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Question 1
1.
Use the Tree Diagram
You can count the end of all the "branches" to find total possibilities. Total number of possible outcomes for sandwich, side and drink options are _______ possibilities.
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Question 5
5.
Work with a partner and discuss/justify your answers.
A fair conducts three obstacle course races.
In how many different orders can ALL the dogs finish in each race?
Race 1 could only have _______ possible results.
Race 2 could only have _______ possible results.
Race 3 could only have _______ possible results.
We can use factorials to find all possible outcomes for multiple events that drop in value as items are selected.
Example
Consider the letters in the word JULY. In how many ways can you arrange all of the letters?
Solution
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Question 6
6.
Consider the letters in the word PENCILS.
In how many ways can you arrange all of the letters?
Total Possibilities = _______
Permutations
A permutation is an arrangement of objects in which order is important.
Example
Ten horses run in a race. In how many different ways can the horses finish first, second, and third?
(Assume there are no ties.)
Solution:
We only need to choose 3, first, second and third. So, 10 x 9 x 8 = 720
(10 choices for 1st) x (9 choices for 2nd) x (8 choices for 3rd) = 720
Example:
10P3 = 10 x 9 x 8 = 720
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Combinations
A combination is a selection of objects in which order is not important.
Example 1:
In a drawing for 3 identical prizes, the order of the winners does not matter. If the prizes were different, then the order would matter. How many different ways can the identical prizes be awarded to 20 contestants?
Solution:
20C3 = 1,140
Example 2:
Count the possible combinations of 2 letters chosen from the list A, B, C, D.
Solution:
List all of the permutations of 2 letters from the list A, B, C, D. Because order is not important in a combination, cross out any duplicate pairs.
There are 6 possible combinations of 2 letters from the list A, B, C, D.
4C2 = 6
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Question 2
2.
Use the Fundamental Counting Principle to find the total number of possible outcomes for sandwich, side and drink options.
3 Events:
(# of Sandwich options) x (# of Side options) x (# of drink options) = Total Possible Outcomes
_______ x_______ x_______ =_______
Question 3
3.
Answer each probability as a fraction and percent, to the nearest tenth.
a. P(Chicken, Chips, Soda) = _______=_______%
b. P(Chicken) = _______ = _______%
c. P(Chips) = _______ = _______%
d. P(Soda) = _______ = _______%
Question 4
4.
Answer each probability as a fraction and percent, to the nearest tenth.
a. P(Chips ∩ French Fries) = _______ = _______%
b. P(Chips ∪ Fruit Cup) = _______ = _______%
c. P(Turkey ∩ Soda) = _______ = _______%
d. P(Grilled Cheese ∪ Fruit Cup) =_______ = _______%
Question 7
7.
Consider the letters in the word
JULY.
In how many ways can you arrange 2 of the letters?
Question 8
8.
Consider the letters in the word PENCILS.
In how many ways can you arrange 3 of the letters?
Total Possibilities = _______
Question 9
9.
You ride on a float with your soccer team in a parade. There are 12 floats in the parade, and their order is chosen at random. Find the probability that your float is first and the float with the school chorus is second.
Total Possibilities:
The total number possible outcomes for 1st and 2nd are _______ outcomes.
Favorable:
How many of those possibilities includes your float first and the float with the school chorus second? _______
So, P(Soccer Team 1st, Chorus 2nd) = _______ (favorable/Total)
Question 10
10.
Consider the letters in the word MARCH. In how many ways can you arrange...
a. All of the letters? _______
b. 3 of the letters? _______
Question 11
11.
There are 14 floats in the parade. Find the probability that the soccer team is first and the chorus is second.
P(Soccer Team 1st, Chorus 2nd) = _______
Question 12
12.
Eight people serve on a committee. In how many different ways can a chairperson, a recorder, and a treasurer be chosen from the committee members?
Question 13
13.
You and your friend are auditioning for a part in the school play. There are 15 people auditioning, and the order of their auditions is chosen at random. Find the probability that your audition is last and your friend’s audition is second to last.
P(You last, friend 2nd last) = _______
Question 14
14.
You order a sandwich at a restaurant. You can choose 2 side dishes from a list of 8. How many combinations of side dishes are possible?
Question 15
15.
Count the possible combinations of 4 letters chosen from the list P, Q, R, S, T, U.
Question 16
16.
You order a sandwich at a restaurant. You can choose 2 side dishes from a list of 8. How many combinations of side dishes are possible?
Total Possibilities = _______
Question 17
17.
You are listening to music. You have time to listen to 3 songs from your playlist of 16 songs. How many combinations of 3 songs are possible?
Question 18
18.
Count the possible combinations of 3 letters chosen from the list A, B, C, D, E.
Question 19
19.
A yearbook editor has selected 14 photos, including one of you and one of your friend, to use in a collage for the yearbook. The photos are placed at random. There is room for 2 photos at the top of the page. What is the probability that your photo and your friend’s photo are the 2 placed at the top of the page?
Total Possibilities:
Does order matter?
The total number of possible outcomes as the number of combinations of 14 photos taken 2 at a time are _______
Favorable:
How many of those possibilities includes your photo and your friend's photo? _______
So, P(your and your friend’s photos are chosen) = _______ (favorable/Total)
Question 20
20.
An art teacher has selected 13 projects, including one of yours and one of your friend’s, to put into a display case in the hallway. The projects are placed at random. There is room for 2 projects in the middle row of the case. What is the probability that your project and your friend’s project are the 2 placed in the middle row?