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More Challenging StuffThe classical method, when all
outcomes are equally likely, involves counting the number of ways something can occur
This section includes techniques for counting the number of results in a series of choices, under several different scenarios
Example● If there are 3 different colors of
paint (red, blue, green) that can be used to paint 2 different types of toy cars (race car, police car), then how many different toys can there be?
● 3 colors … 2 cars … 3 • 2 = 6 different toys
● This can be shown in a table or in a tree diagram
Table A table of the different possibilities
This is a rectangle with 2 rows and 3 columns … 2 • 3 = 6 entries
Tree DiagramA tree diagram of the different
possibilities
Red
Blue
Green
Blue Race Car
Blue Police Car
Green Race Car
Green Police Car
Red Race Car
Red Police Car
Race
Police
Race
Police
Race
Police
PaintCar
Multiplication Rule of CountingThe Multiplication Rule of Counting
applies to this type of situation If a task consists of a sequence of
choices With p selections for the first choice With q selections for the second
choice With r selections for the third choice …
Then the number of different tasks is
p • q • r • …
Example● Example Part A● A child is coloring a picture of a
shirt and pants● There are 5 different colors of
markers● How many ways can this be
colored?● By the multiplication rule5 • 5 = 25
Different Example● Example Part B● A child is coloring a picture of a shirt and
pants● There are 5 different colors of markers● The child wants to use 2 different colors● How many ways can this be colored?● By the multiplication rule5 • 4 = 20
Example continued
● Allowing the same marker to be used twice5 • 5 = 25● Requiring that there be two different
markers5 • 4 = 20● There are 5 selections for the first choice for
both Part A and Part B of this example● But they differ for the second choice … there
are only 4 selections for Part B
Repetition??
● Example continued● Part A, allowing the same marker to be
used twice, is called counting with repetition and has formulas such as
5 • 5 • 5 • …● Part B, requiring that there be two
different markers, is called counting without repetition and has formulas such as
5 • 4 • 3 • …
Calculator Commands While in a
CALCULATOR Page:
Menu, Probability,
We will be using Factorial, permutations, & combinations
Factorial● One way to help write these products is using
the factorial symbol n!n! = n • (n-1) • (n-2) • … • 2 • 1● We start off by saying that0! = 1 and 1! = 1● For example5! = 5 • 4 • 3 • 2 • 1 = 120● Notice how 5! looks like the 5 • 4 • 3 from the
previous example
Permutation (Order Matters)● The problem of choosing one marker out
of 5 and then choosing a second marker out of the 4 remaining is an example of a permutation
● A permutation is an ordered arrangement, in which r different objects are chosen out of n different objects with repetition not allowed
● The number of ways is written nPr
Permutation Formula A mathematical way to write the formula for the
number of permutations is
This is a very convenient mathematical way to write a formula for nPr, but it is not a particularly efficient way to actually compute it
In particular, n! gets rapidly gets very large
Order● For some problems, the order of choice
does not matter● Order matters example
Choosing one person to be the president of a club and another to be the vice-president
Two different roles● Order does not matter example
Choosing two people to go to a meeting The same role
Combination (Order Does Not Matter)● When order does not matter, this is
called a combination● A combination is an
unordered arrangement, in which r different objects are chosen out of n different objects with repetition not allowed
● The number of ways is written nCr
Permutation vs. Combination Comparing the description of a permutation
with the description of a combination
The only difference is whether order matters
Combination Formula Because each combination corresponds
to r! permutations, the formula nCr for the number of combinations is
Example● If there are 8 researchers and 3 of them are to
be chosen to go to a meeting● A combination since order does not matter
● There are 56 different ways that this can be done
Permutation or Combination● Is a problem a permutation or a
combination?● One way to tell
Write down one possible solution (i.e. Roger, Rick, Randy)
Switch the order of two of the elements (i.e. Rick, Roger, Randy)
● Is this the same result? If no – this is a permutation – order matters If yes – this is a combination – order does not
matter
DifferentOur permutation and combination
problems so far assume that all n total items are different
Sometimes we have a permutations but not all of the n items are different
This is a more complicated problemHow many ways are there?
Example● How many ways to put 3 A’s, 2 N’s, and 2
T’s to try to make a seven letter sequence?
____ ____ ____ ____ ____ ____ ____● Each of the blanks can be filled in with
either an A or a N or a T● The three A’s are the same … the two N’s
are the same … the two T’s are the same
Example ContinuedWhere can the A’s go?
There are 7 possible placesAny 3 of them are possibleOrder does not matterSo 7C3 different ways to put in the A’s
Example ContinuedWhere can the N’s go?
There are 4 possible places (since 3 of the 7 have been taken by the A’s already)
Any 2 of them are possible Order does not matter So 4C2 different ways to put in the N’s
And there are 2C2 different ways to put in the T’s
Example Continued● Altogether there are
7C3 • 4C2 • 2C2
different ways● This is
● Notice that the denominator is 3, 2, 2 … the numbers of each letter
Permutation● A permutation example● In a horse racing “Trifecta”, a gambler
must pick which horse comes in first, which second, and which third
● If there are 8 horses in the race, and every order of finish is equally likely, what is the chance that any ticket is a winning ticket?
● Order matters, so this is a permutations problem
Permutation Cont.A permutation example continuedThere are 8P3 permutations of the
order of finish of the horsesThe probability that any one ticket
is a winning ticket is 1 out of 8P3, or 1 out of 336
Combination Example● A combination example● The Powerball lottery consists of choosing 5
numbers out of 55 and then 1 number out of 42
● The grand prize is given out when all 6 numbers are correct
● What is the chance of getting the grand prize?
● Order does not matter, so this is a combinations problem (for the 5 balls)
Combination Cont.● A combination example continued● There are 55C5 combinations of the 5 numbers
● There are 42 possibilities for the last ball, so the probability of the grand prize is 1 out of
which is pretty small
SummaryThe Multiplication Rule counts the
number of possible sequences of itemsPermutations and combinations count
the number of ways of arranging items, with permutations when the order matters and combinations when the order does not matter
Permutations and combinations are used to compute probabilities in the classical method