Transcript Permutations and Combinations

Do Now
• Suppose Shaina’s coach has 4 players in mind for
the first 4 spots in the lineup. Determine the number
of ways to arrange the first four batters.
HW KEY (Pg. 775-776 #18, 19-31)
1)
Dependent, 28/253 or about 11%
23) P(white, white) = 91/276 or about 33%
2)
Dependent, 3/38 or about 8%
24) P(heart or spade) = ½ or 50%
3)
Independent, 4/9 or about 44%
25) P(spade or club) = ½ or about 50%
4)
Independent, 64/289 or about 22%
26) P(queen, then heart) = 4/221 or about 2%
5)
Mutually exclusive; 2/13 or about 15%
27) P(jack, then spade) = 4/221 or about 2%
6)
Mutually exclusive; ½ or about 50%
28) P(five, then red) = 25/663 or about 4%
7)
Not mutually exclusive; 4/13 or about 31%29) P(ace or black) = 7/13 or about 54%
8)
Not mutually exclusive; 4/13 or about 31%30) P(red, red, orange) = 125/5488 or about 2%
19) P(red, blue) = 8/87 or about 9%
31) a) 345
20) P(blue, yellow) = 16/145 or about 11%
b) 159
21) P(yellow, not blue) = 42/145 or a bout 29%
c) 227/345 or about 66%
22) P(red, not yellow) = 17/87 or about 20%
d) 2/23 or about 9%
Permutations
and
Combinations
Sample Space
• The list of all of the choices in a group.
o When the objects are arranged so that the order is
important and every possible order of the objects is
provided, the arrangement is called a permutation.
o When the order is not important, it is called a combination.
Why?
Scheduling Options
• Algebra
• Biology
• Language Arts
• Spanish
• World History
The list shows the classes you plan to take next year.
You wonder how many different ways there are to
arrange your schedule for the first three periods of the
day.
a. Make a tree diagram that lists all of the possibilities
for the three periods. Do not repeat any classes in
each arrangement.
b. How many different choices did you have for the
first period? The second period? The third period?
Definition
• Permutation: an arrangement or
listing in which order is important
• Notation - P(5,3) represents 5 things,
taken 3 at a time.
• Formula – P(n,r ) = n!/(n-r)!
Example 1
• The librarian is placing 6 of 10 magazines on a shelf
in a showcase. How many ways can she arrange
the magazines in the case?
Practice
• A designer has created 15 outfits and needs to
select 10 for a fashion show. How many ways can
the design arrange the outfits for the show?
• Four of ten different fruits are being selected to be
placed in a row in a display window. In how many
ways can they be placed?
Example 2
• How many zip codes can be made from the digits
0, 7, 5, 4, and 6 if each digit is used only once?
Definition
• Sometimes order doesn’t matter!
Example: chocolate, vanilla, and strawberry is the same as
strawberry, chocolate, and vanilla when you order ice
cream.
• Combination: an arrangement or listing where order is
not important
• Notation - C(5,3) represents 5 things, taken 3 at a time
where the order does not matter.
• Formula – C(n, r) = n!/[(n-r)!r!]
Example 3
• How many ways can students choose two flavors of
sherbet from orange, lemon, strawberry, and
raspberry?
• Lucas works part-time at a local department store.
His manager asked him to choose for display 5
different styles of shirts from the wall of the store that
has 8 shirts on it to put in a display. How many ways
can Lucas choose the shirts?
Practice
• In how many ways can you choose two student
council representatives from the students shown?
Jack, Joey, Adra, Aidan, Bryan
• How many ways can a customer choose 3 pizza
toppings from pepperoni, onion, sausage, green
pepper, and mushroom?
Example 4
• A combination lock requires a three-digit code
made up of the digits 0 through 9. No number can
be used more than once.
o How many different arrangements are possible?
o What is the probability that all of the digits are odd?
Practice
• The Spanish club is electing a president, vice
president, secretary, and treasurer. Rebekah and
Lydia are among the nine students who are running.
o How many ways can the Spanish club choose their officers?
o Assuming that the positions are chosen at random, what is the probability
that either Rebekah or Lydia will be chosen as president or vice president?
Homework
• Pg. 768-769 #9, 10, 11, 14, 18, 20-31 All