Download Aim: How do we determine the number of outcomes when order is

Aim: How do we determine the number of
outcomes when order is not an issue?
Do Now:
Ann, Barbara, Carol,
and Dave are the
only members of a
school club. In how
many different ways
can they elect a
president and
treasurer for the
club?
Ann, Barbara, Carol,
and Dave are the
only members of a
school club. In how
many ways can they
choose 2 people to
represent the club at
student council
meetings?
Explain how these situations are different.
Aim: Combinations
Course: Math Lit.
Subsets & Arrangements
A = {a, b, c, d, e}
If order were important
is {a, b} = {b, a} ?
No
If the two elements a and b are selected from A,
then
there is one subset (order not important): {a, b}
there are two arrangements (order important):
{a, b} and {b, a}
A = {Ann, Barbara, Carol, Dave}
order is important
Barbara
Carol
Carol
Dave
Ann
member
of council
president
Aim: Combinations
Barbara
Dave
Carol
Dave
Ann
member
of council
treasurer
Course: Math Lit.
Permutation
Ann, Barbara, Carol, and Dave are the only
members of a school club. In how many
different ways can they elect a president
and treasurer for the club?
4P2 = 4 • 3 = 12
Treasurer.
President
Ann
Barbara
Carol
Dave
Barbara
Carol
Dave
Ann
Carol
Dave
Barbara
Ann
Dave
Barbara
Carol
Ann
Ann & Barbara
Ann & Carol
Ann & Dave
Barbara & Ann
Barbara & Carol
Barbara & Dave
Carol & Barbara
Carol & Ann
Carol & Dave
Dave & Barbara
Dave & Carol
Dave & Ann
Aim:are
Combinations
Course: Math Lit. of
There
12 different arrangements
two people for president and treasurer.
Combination
Ann, Barbara, Carol, and Dave are the only
members of a school club. In how many
ways can they choose 2 people to represent
the club at student council meetings?
1st Person
Ann
Barbara
Carol
Dave
2nd Person
Barbara
Carol
Dave
Ann
Carol
Dave
Barbara
Ann
Dave
Barbara
Carol
Ann
Combinations
There are sixAim:combinations
of
two people that can represent
Ann & Barbara
Ann & Carol
Ann & Dave
Barbara & Ann
Barbara & Carol
Barbara & Dave
Carol & Barbara
Carol & Ann
Carol & Dave
Dave & Barbara
Dave & Carol
Dave & Ann
Course: Math Lit.
Order: Permutation vs. Combination
A selection of objects in which their
order is not important.
When selecting some of the objects in the set:
The number of
combinations of n
objects r at a time
 n  n Pr
n Cr    
 r  r!
6!
6! 720
120
(6  3)! 3!
6
6 P3




 20
6 C3 
3!
3!
3!
6
6
When selecting all
objects in the set:
Pn
n Cn 
n!
n
=1
4!
4! 24
there is only 1
P
24
(4

4)!
Aim:
Combinations
Course: Math Lit.
0!  1 
4 4
C




1 combination!!
4 4
4!
4!
4! 24 24
Combinations
Some Special Relationships
1. For any counting number n, nCn = 1
3C3 =
1
10C10 =
1
2. For any counting number n, nC0 = 1
5C0 =
1
34C0 =
1
3. For whole numbers n and r,
where r < n, nCr = nCn - r
7C3
= 7C7 - 3 = 7C4
23C16 = 23C23 - 16
Aim: Combinations
=
23C7
Course: Math Lit.
Combinations & Pascal’s Triangle
0C0 =
1C0 =
2C 0 =
3C0 =
4C0 =
1
1
2C1 =
3C1 =
1 4C 1 = 4
5C0 = 1 5C1 = 5
1
3
4C2 =
1
C
10
5 2=
1
1
1
1
1
1
7
5
3C2 =
16
3C3 =
4C3 =
2
4
1
4C4 =
1
3
10
5C5 =
1
5
15
35
1
1
4
20
35
3
1
1 10 C
C
5 3=
5 4= 5
10
Aim: Combinations
1
2C2 =
2
6
15
21
1C1 =
3
4
6
1
1
6
21
7
Course: Math Lit.
1
1
1
Combination
Ann, Barbara, Carol, and Dave are the only
members of a school club. In how many
ways can they choose 2 people to represent
the club at student council meetings?
1st Person
2nd Person
Ann
Pr

n C r Barbara
r!
n
Carol
Dave
Aim: Combinations
Barbara
Carol
Dave
Ann
Carol
Dave
Barbara
Ann
Dave
Barbara
Carol
Ann
4C2
Ann & Barbara
Ann & Carol
Ann & Dave
Barbara & Ann
Barbara & Carol
Barbara & Dave
Carol & Barbara
Carol & Ann
Carol & Dave
Dave & Barbara
Dave & Carol
Dave & Ann
Course: Math Lit.
= 4P2 / 2! = 6
Model Problems
Evaluate: 10C3 = 120
8C2 = 28
How many different three-person
committees can be formed from a group of
eight people?
Is order important? NO
8C3
= 56
A committee has 7 men and 5 women. A
subcommittee of 8 is to be formed. Write
an expression for the number of ways the
choice can be made.
12C8
= 495
In general, use permutations where order
is important, and combinations where
Aim: Combinations
Course: Math Lit.
order is not important.
Model Problem
From an urn containing 4 black marbles, 8
blue marbles, and 5 red marbles, in how many
ways can a set of 4 marbles be selected?
Is the order of the 4 marbles important?
NO!
Pr
Combination n C r  C  n, r  
r!
n
17 total marbles
4
17C4 =
5 7
17  16  15  14
4 3  2 1
Aim: Combinations
= 2380
Course: Math Lit.
Model Problem
If nC2 = 15, what is the value of n?
Pr
n Cr 
r!
n(n  1)
n P2



15
C
n 2
2!
21
n
n(n - 1) = 2•15
n2 - n = 30
n2 - n - 30 = 0
(n - 6)(n + 5) = 0
(n - 6) = 0
n=6
6C2
Aim: Combinations
(n + 5) = 0
n = -5
= 15
Course: Math Lit.
Fundamental and Combinations
A committee of five is chosen from five
mathematicians and six economists. How
many different committees are possible if
the committee must include two
mathematicians and three economists?
mathematicians:
5C2
economists:
6C3
.
Aim: Combinations
= 10 · 20 = 200
Course: Math Lit.
Model Problem
The US Senate of the 104th Congress
consisted of 54 Republicans and 46
Democrats. How many committees can be
formed if each committee must have 3
Republicans and 2 Democrats?
Republicans:
54C3
Democrats:
46C2
.
= 24,804 · 1035
= 25,672,140
Aim: Combinations
Course: Math Lit.
Model Problems
There are 10 boys and 20 girls in a class.
Find the number of ways a team of 3
students can be selected to work on a project
if the team consists of:
A. Any 3 students
30C3
B. 1 boy and 2 girls
10C1
= 4060
•
20C2
= 10 • 190 = 1910
C. 3 girls
10C0
•
20C3
2 girls
D. At least 2 girls
Aim: Combinations
10C1
•
20C2
= 1140
3 girls
+
10C0
•
20C3
= 1910 + 1140
Course: =
Math3040
Lit.
Model Problem
In how many ways can 6 marbles be
distributed in 3 boxes so that 3 marbles are
in the first box, 2 in the second, and 1 in the
third
Box 1
6C3
20
Box 2
•
•
Aim: Combinations
3C2
3
Box 3
•
•
1C1
1
Course: Math Lit.
= 60
Model Problem
Find the number of ways to select 5-card
hands from a standard deck so that each
hand contains at most 2 aces.
at most 2 aces
Means that the hand could have 0, 1 or 2 aces
W/ no Aces
4C0
•
48C5
= 1712304
W/ 1 Aces
4C1
•
48C4
= 778320
W/ 2 Aces
4C2
•
48C3
+ = 103776
Aim: Combinations Complete the
Choose Aces
5-card hand
= 2594400
Course: Math Lit.
Aim: How do we determine the number of
outcomes when order is not an issue?
Do Now:
In the “Pick Four” Lottery, you create a 4-digit
number using the numbers 1, 2, 3, 4, 5, and 6.
If you play the game “straight”, you win if the
winning lottery number matches your selection
exactly. How many different arrangements are
possible if you bet the game “straight”?
6P4
Aim: Combinations
= 360
Course: Math Lit.
Model Problems
In the “Pick Four” Lottery, you create a 4-digit
number using the numbers 1, 2, 3, 4, 5, and 6.
If you play the game “straight”, you win if the
winning lottery number matches your selection
exactly. How many different arrangements are
possible if you bet the game “straight”?
6P4
= 360
If you choose you may, you may play the game
“boxed”. This means that as long as the same
four numbers are chosen, regardless of order,
you win. How many possible combinations are
Course: Math Lit.
possible? Aim: Combinations6C4 = 15
Model Problem
How many different 4-member committees
can be formed from a group of 10 people if
Tony, 1 of the 10 must:
9C3 = 84
A. Always on the committee
1 • 9C3
TONY
IS A
MUST!
Aim: Combinations
= 84
AFTER TONY IS PLACED ON
THE COMMITTEE, THERE
ARE 3 PLACES LEFT FOR
THE OTHER 9 PEOPLE
Course: Math Lit.
Model Problem
How many different 4-member committees
can be formed from a group of 10 people if
Tony, 1 of the 10 must:
B. Never be on the committee
There are now only 9 possible
members for the 4-member committee
9C4
Aim: Combinations
= 126
Course: Math Lit.
Counting Techniques
Tree Diagram Fundamental Combinations Permutations
Counting
Principle
Use this to
Counts total
Repetitions not allowed
handle
number of
inconsistencies separate tasks
most tedious,
Arrangements
Repetitions
Subsets
use when all
allowed
else fails
total number
Order does
Order
of ways a task not matter
matters
can be
n!
n!
performed n C r 
n Pr 
r ! n  r  !
 n  r !
m · n · o · p ···
Aim: Combinations
Course: Math Lit.
Model Problems
Sets of 2 letters are chosen from the English
alphabet. Find the number of 2-letter
sets possible if the set:
a. cannot have a vowel
b. cannot have a consonant
c. must have at most 1 vowel
d. must have a vowel and a consonant
Aim: Combinations
Course: Math Lit.
Model Problems
Find the number of ways a coach can select
her starting basketball team from a group
of 12 players, 8 boys and 4 girls, if the
positions to be played are not taken into
account, and if:
a. Sally, 1 of the players is always on the
team
b. Ed, 1 of the players is never on the team.
c. both Sally and Ed are not on the team
d. either Sally or Ed, but not both, is on the
team.
Aim: Combinations
Course: Math Lit.
Model Problems
Sets of 4 letters are chosen from the English
alphabet. Find the number of 4-letter lets
possible if there must be the same
number of vowels and consonants, and if:
a. A is always included
b. M is always included
c. E is never included
d. Q is never included
Aim: Combinations
Course: Math Lit.