4. The student council of 32 members must choose a president, a vice president, a secretary, and a treasurer. How many combinations of officers could there be? 3. How many different ways can 10 students line up for lunch? 1.Christian is going to dinner at Olive Garden. He is allowed to choose one of four pastas, one of three sauces, and one of three meats. How many different pasta dished could Christian make? 2. How would the number of combinations change if they were out of one of the sauces?
Transcript
4. The student council of 32 members must choose a president, a vice president, a secretary, and a treasurer. How many combinations of officers could there be?
3. How many different ways can 10 students line up for lunch?
1. Christian is going to dinner at Olive Garden. He is allowed to choose one of four pastas, one of three sauces, and one of three meats. How many different pasta dished could Christian make?
2. How would the number of combinations change if they were out of one of the sauces?
Let’s say there are 15 student council members named: Bruce, Shareeka, Yasmine, Jalen, Greg,
Stephon, Mikey, Bria, Joseph, Ekure, Anthony, Ar-Keno, Brittany, Tiffany, and Amber. If Shareeka, Tiffany, Yasmine, and Bruce are elected, would
the order in which they are chosen matter?
President Vice President Secretary Treasurer
Although the same individual students are listed in each example above, the listings are not the same. Each
listing indicates a different student holding each office. Therefore we must conclude that the order in which
they are chosen matters.
Shareeka Tiffany Yasmine Bruce
the same as…
Bruce Yasmine Tiffany Shareeka?
IS…
Permutation
Notation
Permutation
When deciding who goes 1st, 2nd, etc., order is important.
*Note if n = r then n Pr = n !
A permutation is an arrangement or listing of objects in a specific order.
The order of the arrangement is very important!!
The notation for a permutation: n Pr = n is the total number of objects r is the number of objects selected (wanted)
!
( )!
n
n r
PermutationsSimplify each expression.
a. 12P2
b. 10P4
c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible?
12 • 11 = 132
10 • 9 • 8 • 7 = 5,040
= 20P4 = 20 • 19 • 18 • 17 = 116,280
Permutation Example:
Four runners are needed to run the 400 meter relay. How many different arrangements are there for the four runners to run the 4 legs
of the race?
The team has 8 sprinters. Four runners are needed to run the 400 meter relay. How
many different arrangements are there for the four runners, choosing from the 8 team
members, to run the 4 legs of the race?
24
1680
24
1.Bugs Bunny, King Tut, Kevin Jerome, and Daffy Duck are going to the movies (they are best friends). How many different ways can they sit in seats A, B, C, and D below?
2. Coach Hamilton is picking a captain and co-captain from her 15 players. How many possibilities does she have if they are all equally likely?
A B C D
210
Combinations
A selection of objects in which order is not important.
Example – 8 people pair up to do an assignment. How many different pairs are there?
CombinationsAB AC AD AE AF AG AH
BA BC BD BE BF BG BH
CA CB CD CE CF CG CH
DA DB DC DE DF DG DH
EA EB EC ED EF EG EH
FA FB FC FD FE FG FH
GA GB GC GD GE GF GH
HA HB HC HD HE HF HG
• The number of r-combinations of a set with n elements,
• where n is a positive integer and • r is a positive integer less than n, • i.e. the number of combinations of
r objects from n unlike objects is
!
! !n r
nC
r n r
Combinations
Example 1
How many different
ways are there to
select two class
representatives
from a class of 20
students?
!
! !n r
nC
r n r
Solution
• The answer is given by the number
of 2-combinations of a set with 20
elements.• The number of such combinations
is
20 2
20!190
2!18!C
Example 2
From a class of 24, Mrs. Shaffer is randomly selecting 3 to help Mrs. Benson with a project. How many
combinations are possible?
24 3
24!2024
3! 21!C
Your turn!For your school pictures, you can choose 4 backgrounds from a list of 10. How many combinations of backdrops are possible?
10 4
10!210
4! 6!C
Your turn!Coach Lynch randomly selects 3 people out of his class of 20 to go help him get ready for a lacrosse match. How many possibilities of people does he have?
20 3
20!1140
17! 3!C
To Sum it Up:
!
! !n r
nC
r n r
!
!n r
nP
n r
"My fruit salad is a combination of
apples, grapes and bananas"
the order doesn't matter
"The combination to
the safe is 472".
the order DOES matter
A Permutation is an ordered
Combination.
Both are counting principles
that tell you the total
number of possible
outcomes
Clarification on Combinations and Permutations
• "My fruit salad is a combination of apples, grapes and bananas"
We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.
Clarification on Combinations and Permutations
• "The combination to the safe was 472".
Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.
To sum it up…
• If the order doesn't matter, it is a Combination.
• If the order does matter it is a Permutation.
A Permutation is an ordered Combination.
How many ways???
Fill in the blanks….
1. A permutation is an _________________ of objects in which order ___________ matter.
2. A combination is an _________________ of objects in which order ____________ matter.
