Transcript combination

pencil, highlighter, GP notebook,
textbook, calculator
Evaluate the following WITHOUT using a calculator.
(i.e. do them by hand until the last step!)
11!
11!
+1
+1
a) 11P6 
b) 11C6 
(11  6)!6!
(11  6)!
11!
11!


5!6!
5!
11 10  9  8  7  6!
11 10  9  8  7  6  5!
+1

+1

5!6!
5!
11 10  9  8  7
= 332,640 +2

120
+2
=
462
total:
8
1. DO NOT SOLVE! Just determine whether you would use a
permutation or a combination for each of the following
a) Nathan has a dozen eggs. He wants to decorate 4 eggs for
an art project.
combination
b) Mr. Gunther is lining up 12 kindergarten students for a
performance.
permutation
c) Rachel has 10 valuable baseball cards. She wants to select 2
of them to sell online.
combination
d) Don has 5 soccer trophies to line up on the mantel of his
fireplace.
permutation
e) Bree has to select 5 photos from a box containing 25 photos
to use in the yearbook.
combination
2. Solve each problem.
a) How many ways can Fred order a Coldstone sundae with
his favorite flavor of ice cream if he chooses 4 of the 13
available toppings?
Order does not matter...
13C4
= 715
Combination
b) Joe is putting together a box of 6 pieces of See’s Candy.
How many boxes are possible if there are 24 pieces Joe likes,
and he gets no repeats?
Order does not matter... Combination
24C6
= 134,596
c) A reading list for a course in world literature has 25 books on
it. In how many ways can you choose 3 books to read?
Order does not matter... Combination
25C3
= 2300
d) How many different committees of 12 students can be
chosen for 3 seats in ASB?
Order does not matter... Combination
12C3
= 220
e) Eight sorority sisters are running for the advisory council.
 In how many ways can 4 girls be chosen?
Order does not matter... Combination
8C4
=
70
 In how many ways can a president, vice–president, treasurer,
and secretary be selected from the 8 girls?
Order DOES matter...
8P4
=
1680
no repeats… Permutation
f) In how many ways can the B column of a Bingo card be
arranged with 5 of the numbers from 1 – 15?
Order DOES matter...
no repeats…
Permutation
15P5
= 360,360
g) Five friends are assigned the 5 middle seats on a 747. How
many ways can they be arranged?
Order DOES matter... no repeats…
Permutation
5 •
4
• 3
• 2
•
1
= 5! = 5P5 = 120
IC – 68
Is a combination lock correctly named? Andrea
bought a standard dial lock. The numbers are from 0
to 39. A combination is any three numbers… but
order matters. Therefore, it is a ___________
permutation lock.
_____
permutations are possible if:
How many three–number “combinations”
a) no number is repeated?
40!
40!

P

 40  39  38  59,280
40 3
(40  3)! 37!
How many three–number “combinations” are possible if:
b) repeats are allowed?
40
40
40
= 403
= 64000
Does order matter?
NO
Are repeats allowed?
NO
Combination
nCr
YES
YES
Are repeats allowed?
NO
Permutation
nPr
YES
Decision Chart
nr
Joaquin is getting a new locker at school and the first thing he
must do is decide on a new combination. The three number
combination can be picked from the numbers 0 – 21. How
many different locker combinations can Joaquin choose if:
a) no number is repeated?
order matters, no repeats
Permutation
22P3
22!
22 P3 
(22  3)!
22!
19!
 22  21 20

 9240
b) repeats are allowed?
order matters, repeats allowed
Decision chart/Exponent
223
= 10648
IC – 69
Replacement
a) A pollster has the names of 8 people available to answer her
questions. She must select 3 of the people to interview.
When she selects interviewees, does the order of
no
respondents matter? ______
no
Can the respondents be re-interviewed (repeated)? ____
How many ways can she select the respondents?
Use Combination
8!
876
8!


 4  7  2  56
8 C3 
(8  3)!3! 5!3! 3  2  1
b) The track team has 6 runners in a race. The coach selects
2 runners to run in the first heat. In how many ways can the
runners be selected?
Does order matter?
no
Can the choices be repeated? no
Use Combination
6 C2 
6!
6!
65


 3  5  15
(6  2)!2!
4!2!
2 1
IC – 70
Replacement
At Pete’s Perfect Pizza, all pizzas come with sauce and
cheese. There are 12 toppings available (onions,
pepperoni, anchovies, sausage, peppers, etc).
a) How many different two topping pizzas are available (with
no repeated toppings)? (First decide: is this a permutation
or combination???) Use Combination
12!
12  11
12!


 6  11  66
12 C 2 
2 1
(12  2)!2! 10!2!
b) How many three topping pizzas are available (with no
Use Combination
repeated toppings)?
12! 12  11 10
12


 2  11 10  220
12 C 3 
3  2 1
(12  3)!3! 9!3!
Clear your desk except for a pencil,
highlighter, and a calculator!
After the quiz,
work on the rest
of the assignment.
IC – 71
At Burger King®, you can “have it your way” by
ordering your burger with or without mustard,
ketchup, mayonnaise, lettuce, tomato, pickles,
cheese, and onion.
a) How many ways can you “have it your way”?
8 toppings, two choices each
2 2 2 2
2
2 2 2
= 28
= 256
explanation
b) How many ways can the condiments be arranged if you
decided to order everything on your burger?
order matters; no repeats
8P8
= 8! = 40320
IC – 71
At Burger King®, you can “have it your way” by
ordering your burger with or without mustard,
ketchup, mayonnaise, lettuce, tomato, pickles,
cheese, and onion.
c) How many combinations of any three condiments can you
order on your burger? order doesn’t matter, no repetition
8!
8!
876


 56
8C3 
(8  3)!3! 5!3! 3  2  1
d) Suppose you only want cheese, ketchup and mustard. How
many different ways can these condiments be placed on your
burger?
order matters, no repeats
3P3
= 3! = 6
IC – 72
Replacement
Answer the following regarding a 10-question true/false quiz.
a) How many true/false arrangements are possible?
10 questions, two choices
210 = 1024
2
2 2
2
2
2 2 2
2 2
b) What is the probability of guessing all questions correctly?
1
P (all correct) = 1024
one way to do that
out of 1024 ways
to answer
IC – 72
Replacement
Answer the following regarding a 10-question true/false quiz.
c) What is the probability that none of your guesses are correct?
1 one way to do that
P (all wrong) = 1024 out of 1024 ways
to answer
d) If you guess on all the questions, how many would you
expect to guess correct?
1
P(one right) =
2
E(test) =
1
(10) = 5 questions
2
IC – 73
Replacement
Answer the following regarding an 8-question
multiple choice quiz, with choices A, B, C, and D.
a) What is the probability of guessing all questions correctly?
1
P(right) =
4
3
P(wrong) =
4
b) How many arrangements of correct answers are possible?
48
= 65536
4
4 4
4
4
4 4 4
c) What is the probability of guessing all questions correctly?
1
P(all right) =
65536
IC – 73
Replacement
Answer the following regarding an 8-question
multiple choice quiz, with choices A, B, C, and D.
d) What is the probability of guessing none correct?
8
3   6561

P (none right) =  
 0.10
 4  65536
e) What is the probability of guessing at least one question
correctly?
P (at least 1 correct) = 1 – P (none correct)
 1 0.10
 0.90
f) If you guess on all the questions, how many would you
expect to guess correct?
1
(8)  2 correct
E(Test)
=
1
4
P(right) =
4
Finish the assignment:
IC 77, 78, 80, 81, and worksheet
Optional
just for “fun” activity
Flip to the back of today’s worksheets:
FOR FUN: Let’s go back to the quizzes in IC – 72 and IC – 73.
IC – 72: Choose either TRUE or FALSE for questions 1 – 10.
Fill in your answers below, then make a histogram of the
data from the class.
It’s time to grade the quiz!!! Take out a red pen.
The answers are…
T 2. __
F 3. __
F 4. __
F 5. __
F 6. __
T 7. __
T 8. __
F 9. __
F 10. __
T
1. __
Write the number correct out of 10 on your worksheet.
Let’s make a bar graph of the results of our class.
7
6
# of
students
5
4
3
2
1
1
2
6 7
3 4
5
# of correct answers.
8
9
How does the class data compare to the expected value?
10
IC – 73: Choose A, B, C, or D for questions 1 – 8. Fill in your
answers below, then make a histogram of the data from
the class.
It’s time to grade the quiz!!! Take out a red pen.
The answers are…
C 2. __
D 3. __
C 4. __
D 5. __
A 6. __
C 7. __
A 8. __
A
1. __
Write the number correct out of 8 on your bellwork.
Let’s make a bar graph of the results of our class.
7
6
# of
students
5
4
3
2
1
1
2
6 7
3 4
5
# of correct answers.
8
How does the class data compare to the expected value?
Would you like mustard, ketchup,
mayonnaise, lettuce, tomato,
pickles, cheese, and onion?
mustard
2
yes
no
ketchup
x2
yes
no
yes
no
x2
mayonnaise
.
.
.
Return
yes
no
.
.
.
.
and on and on…
.
28 = 256
.
old bellwork
pencil, highlighter, GP notebook,
textbook, calculator
a) On the way to the movies you and your 7 friends get ice
cream. The place you are going simply mixes up the
ingredients before serving them in a cup. How many
different kinds of different ice cream cups can you have if
you can only get 3 scoops and there are 12 different flavors
(no repetition allowed)?
b) 8 people go to the movies. The people include a pair of
conjoined twins, one of their dates, two other friends with
both of their dates, and you are a loner. (So sad , but you
are not the only one! The other conjoined twin is a loner
too… sort of.) How many different ways are there to
arrange yourselves?
a) On the way to the movies you and your 7 friends get ice
cream. The place you are going simply mixes up the
ingredients before serving them in a cup. How many
different kinds of different ice cream cups can you have if
you can only get 3 scoops and there are 12 different flavors
(no repetition allowed)?
Does order matter?
NO
Can I repeat myself?
NO
Combination
nCr
YES
YES
Can I repeat myself?
NO
Permutation
nPr
YES
Decision Chart
nr
a) On the way to the movies you and your 7 friends get ice
cream. The place you are going simply mixes up the
ingredients before serving them in a cup. How many
different kinds of different ice cream cups can you have if
you can only get 3 scoops and there are 12 different flavors
(no repetition allowed)?
n!
n Cr 
(n  r)! r!
12C3
12!
12!
12  11 10


 4  11 5  220
12 C 3 
3  2 1
(12  3)!3! 9!3!
same thing
12
11
3!
10
order doesn’t matter, so
divide by the arrangements
b) 8 people go to the movies. The people include a pair of
conjoined twins, one of their dates, two other friends with
both of their dates, and you are a loner. (So sad , but you
are not the only one! The other conjoined twin is a loner
too… sort of.) How many different ways are there to
arrange yourselves?


How many groups are we arranging? 4 groups
Only two groups can change up internally.
4! 2! 2! = 96