**Unformatted text preview: **Section 3.4
Additional Topics in Probability and
Counting 1 Section 3.4 Objectives
Determine the number of ways a group of objects can be arranged in order
Determine the number of ways to choose
several objects from a group without regard
to order
Use the counting principles to find
probabilities 2 Permutations
Permutation
An ordered arrangement of objects
The number of different permutations of n
distinct objects is n! (n factorial)
n! = n∙(n – 1)∙(n – 2)∙(n – 3)∙ ∙ ∙3∙2 ∙1
0! = 1
Examples: 6! = 6∙5∙4∙3∙2∙1 = 720 4! = 4∙3∙2∙1 = 24 3 Example: Permutation of n
Objects
The objective of a 9 x 9 Sudoku number
puzzle is to fill the grid so that each row,
each column, and each 3 x 3 grid contain
the digits 1 to 9. How many different ways
can the first row of a blank 9 x 9 Sudoku
grid be filled? Solution:
The number of permutations is
9!= 9∙8∙7∙6∙5∙4∙3∙2∙1 = 362,880 ways
4 Permutations
Permutation of n objects taken r at a time
The number of different permutations of n
distinct objects taken r at a time
■ 5 n!
where r ≤
n Pr ( n r )! n Example: Finding nPr
Find the number of ways of forming three-digit
codes in which no digit is repeated. Solution:
• You need to select 3 digits from a group of 10
• n = 10, r = 3
10!
10! 10 P3 (10 3)!
7!
10 9 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 720 ways
6 Example: Finding nPr
Forty-three race cars started the 2007 Daytona 500. How
many ways can the cars finish first, second, and third? Solution:
• You need to select 3 cars from a group of 43
• n = 43, r = 3
43!
43! 43 P3 (43 3)! 40! 43 42 41 74, 046 ways
7 Distinguishable Permutations
Distinguishable Permutations
The number of distinguishable permutations of n
objects where n1 are of one type, n2 are of another
type, and so on
■ n!
n1 !n2 ! n3 ! n
k ! where n1 + n2 + n3 +∙∙∙+ nk = n 8 Example: Distinguishable
Permutations A building contractor is planning to develop a subdivision that consists
of 6 one-story houses, 4 two-story houses, and 2 split-level houses. In
how many distinguishable ways can the houses be arranged? Solution:
• There are 12 houses in the subdivision
• n = 12, n1 = 6, n2 = 4, n3 = 2
12!
6!4! 2! 13, 860 distinguishable ways
9 Combinations
Combination of n objects taken r at a time
A selection of r objects from a group of n
objects without regard to order
■ 10 n Cr n! ( n r )! r ! Example: Combinations
A state’s department of transportation plans to
develop a new section of interstate highway and
receives 16 bids for the project. The state plans to
hire four of the bidding companies. How many
different combinations of four companies can be
selected from the 16 bidding companies? Solution:
• You need to select 4 companies from a group of 16
• n = 16, r = 4
• Order is not important
11 Solution: Combinations
16!
16 C4 (16 4)!4!
16! 12!4!
16 15 14 13 12! 12!4 3 2 1 1820 different combinations 12 Example: Finding
Probabilities
A student advisory board consists of 17 members.
Three members serve as the board’s chair,
secretary, and webmaster. Each member is
equally likely to serve any of the positions. What is
the probability of selecting at random the three
members that hold each position? 13 Solution: Finding Probabilities
There is only one favorable outcome
There are 17!
17 P3 (17 3)!
17! 17 16 15 14! 4080 ways the three positions can be filled 1
P( selecting the 3 members ) 0.0002
4080
14 Example: Finding
Probabilities
You have 11 letters consisting of one M, four Is,
four Ss, and two Ps. If the letters are randomly
arranged in order, what is the probability that
the arrangement spells the word Mississippi? 15 Solution: Finding Probabilities
There is only one favorable outcome
There are 11! 34, 650
1!4! 4! 2! 11 letters with 1,4,4,
and 2 like letters distinguishable permutations of the given letters 1
P( Mississippi ) 0.000029
34650
16 Example: Finding
Probabilities
A food manufacturer is analyzing a sample of 400
corn kernels for the presence of a toxin. In this
sample, three kernels have dangerously high levels of
the toxin. If four kernels are randomly selected from
the sample, what is the probability that exactly one
kernel contains a dangerously high level of the toxin? 17 Solution: Finding Probabilities
The possible number of ways of choosing one toxic kernel out of three toxic kernels is
3C 1 = 3
The possible number of ways of choosing
three nontoxic kernels from 397 nontoxic
kernels is
397 C3 = 10,349,790 Using the Multiplication Rule, the number of ways of choosing one toxic kernel and three
nontoxic kernels is
C1 ∙ 3 18 397 C3 = 3 ∙ 10,349,790 3 = 31,049,370 Solution: Finding Probabilities
The number of possible ways of choosing 4 kernels from 400 kernels is
400C4 = 1,050,739,900
The probability of selecting exactly 1 toxic
kernel is C1 397 C3
P(1 toxic kernel ) 400 C4
3 31, 049,370 0.0296
1, 050, 739,900
19 Section 3.4 Summary
Determined the number of ways a group of objects can be arranged in order
Determined the number of ways to choose
several objects from a group without regard
to order
Used the counting principles to find
probabilities 20 ...

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- Fall '17
- CALLHAN
- Group Theory, Probability, ways