10. Permutations and Combinations in Probability
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Chapter 6
10.
Permutations and Combinations in Probability
Understanding permutations and combinations is pivotal for calculating probabilities, especially in compound events. The lesson delves into the distinction between permutations, where the order of arrangement matters, and combinations, where the order is inconsequential. For instance, when considering the arrangement of letters in a word or the order of runners in a race, permutations come into play. On the other hand, combinations are used when selecting a subset, like choosing books from a list. The lesson also touches upon real-world scenarios, such as determining the likelihood of students sharing the same birthday. Grasping these mathematical concepts can significantly aid in solving complex problems in various fields.
Lesson Settings & Tools
| | 10 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Permutations and Combinations in Probability
Slide of 10
In this lesson, the concepts of permutation and combination will be introduced and connected to the computation of probabilities of compound events.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Creating Arrangements
Vincenzo is playing with the following letters.
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Discussion
Defining Permutations and the Permutation Formula
Many situations involve the rearrangement of a specific set of objects. These are called permutation problems. Below, the definition of permutation and its corresponding formula are discussed.
Concept
Permutation
A permutation is an arrangement of objects in which the order is important. For example, consider constructing a number using only the digits 4, 5, and 6 without repetitions. Any of the three digits can be picked for the first position, leaving two choices for the second position, then only one choice for the third position.
In this case, there are six possible permutations. 456 465 546 564 645 654 Although all these numbers are formed with the same three digits, the order in which the digits appear affects the number produced. Each different order of the digits creates a different number. The number of permutations can be calculated by using the Fundamental Counting Principle.
Number of permutations&=3*2*1 &⇕ Number of permutations&=6Rule
Permutation Formula
The number of permutations of n different objects arranged r at a time — denoted as _nP_r — is given by the following formula.
_nP_r=n!/(n-r)!, r≤ n
The exclamation sign in the formula indicates that the factorial of a value must be calculated. As a direct consequence, since 0!=1, when n=r the number of permutations is given by the factorial of n.
_nP_n=n!
An alternative notation for _nP_r is P(n,r).
Proof
Permutation FormulaThe formula can be proven by using the Fundamental Counting Principle. In an arrangement with r elements, there are n choices for the first element, n-1 choices for the second element, n-2 choices for the third element, and so on.
| Position | Number of Choices |
|---|---|
| 1 | n |
| 2 | n-1 |
| 3 | n-2 |
| ... | ... |
| r | (n-r+1) |
By the Fundamental Counting Principle, the product of the choices for each element is equal to the number of different arrangements of n objects chosen r at a time. _nP_r=n(n-1)...(n-r+1) The right-hand side of this equation consists of the first r factors of the factorial of n. The Multiplication Property of Equality can be used to multiply both sides by the last n-r factors of the factorial of n. The product of these factors can be written as (n-r)!
It is important to remember how to write (n-r)! as a product. (n-r)! = (n-r)* (n-r-1) * ... 2 * 1 This expression will be substituted for (n-r)! on the right-hand side of the equation.
_nP_r(n-r)!=n(n-1)...(n-r+1)(n-r)!
Write as a product
_nP_r (n-r)!=n(n-1)...(n-r+1)(n-r)(n-r-1)... (2)(1)
Write as a factorial
_nP_r (n-r)!=n!
DivEqn
.LHS /(n-r)!.=.RHS /(n-r)!.
_nP_r =n!/(n-r)!
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Example
Investigating Permutations in Real-Life Situations
The following cities are the ten most visited cities in Europe.
| Rank | City |
|---|---|
| 1 | London, UK |
| 2 | Paris, France |
| 3 | Istanbul, Turkey |
| 4 | Antalya, Turkey |
| 5 | Rome, Italy |
| 6 | Prague, Czech Republic |
| 7 | Amsterdam, Netherlands |
| 8 | Barcelona, Spain |
| 9 | Vienna, Austria |
| 10 | Milan, Italy |
a Dominika and her friend Heichi are planning to go to Europe next summer. How many different ways can they arrange their trip to see all ten cities?
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b Suppose that they can only visit 3 of the ten places. In how many ways can they do it?
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Hint
a The number of permutations of n out of n is given by the factorial of n.
b Consider the permutation formula for r objects out of n.
Solution
a Because the order in which the cities will be visited is important, the problem can be solved by using permutations. The number of permutations when taking n items out of n is given by the factorial of n.
Therefore, Dominika and Heichi have 3 628 800 different ways to visit all ten most visited cities of Europe.
_nP_n=n! In this case, since there are ten cities to be visited, the factorial of 10 needs to be calculated.
_nP_n=n!
Substitute
n= 10
_(10)P_(10)= 10!
Write as a product
_(10)P_(10)=10*9*8*8*7*6*5*4*3*2*1
Multiply
Multiply
_(10)P_(10)=3 628 800
b Now suppose that Heichi and Dominika will visit only three places. Since the order of the cities they visit is important, permutations can be used again. The number of permutations when taking r items out of n is given by the following formula. _nP_r=n!/(n-r)!
There are 720 ways of visit 3 of the ten cities.
Of the 10 places, only 3 can be visited. Therefore, the number of permutations of 3 out of 10 needs to be calculated.
_nP_r=n!/(n-r)!
SubstituteII
n= 10, r= 3
_(10)P_3=10!/( 10- 3)!
▼
Evaluate right-hand side
SubTerm
Subtract term
_(10)P_3=10!/7!
Write as a product
_(10)P_3=10*9*8*7!/7!
CancelCommonFac
Cancel out common factors
_(10)P_3=10*9*8*7!/7!
SimpQuot
Simplify quotient
_(10)P_3=10*9*8/1
DivByOne
a/1=a
_(10)P_3=10*9*8
Multiply
Multiply
_(10)P_3=720
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Example
Finding Probabilities Using Permutations
In the 2020 Olympic Games, the competitors of the men's 100 meter freestyle swimming finals came from the following countries.
| Men’s 100 Meter Freestyle Swimming Finals | |
|---|---|
| United States | Australia |
| Russia | France |
| South Korea | Italy |
| Hungary | Romania |
a If there were no ties, in how many different ways could the gold, silver, and bronze medals have been awarded?
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b If all athletes have the same athletic ability, what is the probability that the Italian swimmer wins the gold medal, the French swimmer the silver medal, and the Australian swimmer the bronze medal? Approximate the answer to three decimals.
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Solution
a Because the order in which the medals are awarded is essential, the problem can be solved by using permutations. The number of permutations when taking r items out of n is given by the following formula.
_nP_r=n!/(n-r)! Of the 8 swimmers, only 3 can be on the podium and be awarded medals. Therefore, the number of permutations of 3 out of 8 needs to be calculated.
_nP_r=n!/(n-r)!
SubstituteII
n= 8, r= 3
_8P_3=8!/( 8- 3)!
▼
Evaluate right-hand side
SubTerm
Subtract term
_8P_3=8!/5!
Write as a product
_8P_3=8*7*6*5!/5!
CancelCommonFac
Cancel out common factors
_8P_3=8*7*6*5!/5!
SimpQuot
Simplify quotient
_8P_3=8*7*6/1
DivByOne
a/1=a
_8P_3=8*7*6
Multiply
Multiply
_8P_3=336
There are 336 different ways in which the gold, silver, and bronze medals can be awarded.
b Recall that the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
The number of favorable outcomes is the number of ways the Italian, French, and Australian athletes can win the gold, silver, and bronze medals, respectively. Although there is only one way for the order of the first three positions, there are several ways for the order of the remaining five positions. All these are favorable outcomes.
| Example Favorable Outcomes | ||
|---|---|---|
| Italy | Italy | Italy |
| France | France | France |
| Australia | Australia | Australia |
| United States | Hungary | South Korea |
| Russia | Romania | Russia |
| South Korea | Russia | United States |
| Hungary | United States | Hungary |
| Romania | South Korea | Romania |
If the medals are awarded to the Italian, French, and Australian swimmers in this order, then the number of permutations of the last five positions must be calculated. Favorable Outcomes _5P_5=5! ⇔ _5P_5= 120 The number of possible outcomes is the number of all possible ways in which the medals can be awarded. This is found by calculating the permutations of 8 swimmers taken from a group of 8. Possible Outcomes _8P_8=8! ⇔ _8P_8= 40 320 With this information, the probability that the Italian swimmer wins the golden medal, the French swimmer the silver medal, and the Australian swimmer the bronze medal can be calculated.
P(A)=Number of favorable outcomes/Number of possible outcomes
SubstituteValues
Substitute values
P(A)=120/40 320
ReduceFrac
a/b=.a /120./.b /120.
P(A)=1/336
FracToDiv
a/b=a÷ b
P(A)=0.002976...
RoundDec
Round to 3 decimal place(s)
P(A)≈ 0.003
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Discussion
Defining Combinations and the Combination Formula
In other situations, only the selected objects are important, not the order in which they come. These problems are called combination problems. Below, the definition of combination and its corresponding formula are developed.
Concept
Combination
A combination is a selection of objects in which the order is not important. Combinations focus on the selected objects. For example, consider choosing two different ingredients for a salad from five unique options in a salad bar.
Rule
Combination Formula
The number of combinations of n different objects taken r at a time — denoted as _nC_r — is given by the following formula.
_nC_r=n!/r!(n-r)!, r≤ n
The exclamation mark in the formula indicates that the factorial of the value should be calculated. As a direct consequence of the above formula, since 0!=1, when n=r the number of combinations is 1.
_nC_n=1
An alternative notation for _cC_r is C(n,r).
Proof
The formula can be proven by using the Permutation Formulas.
_nP_r=n!/(n-r)! and _rP_r=r!
Let _nC_r be the number of combinations of n objects chosen r at a time. By the Fundamental Counting Principle, the product of _nC_r by _rP_r equals the number of permutations of r objects out of n.
_nC_r* _rP_r = _nP_r ⇓ _nC_r* r!= n!/(n-r)!
Finally, by applying the Division Property of Equality, the Combination Formula is obtained.
_nC_r* r!= n!/(n-r)! ⇕ _nC_r=n!/r!(n-r)!
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Example
Investigating Combinations in Real-Life Situations
Kriz is going on vacation next month and wants to pack 4 books from their must-read list. Each of the books belongs to one of the following genres.
| Kriz’s List of Books By Genres | |
|---|---|
| Fantasy | Romance |
| Mystery | Fiction |
| Biography | Graphic Novel |
| Drama | History |
| Western | Poetry |
In how many ways can they select 4 different books?
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Hint
The order in which the books are selected is not crucial.
Solution
As long as 4 books are selected, the order is not important. Therefore, the different ways in which Kriz can select 4 books can be found by using combinations. The number of combinations when selecting r items out of n is given by the following formula.
_nC_r=n!/r!(n-r)!
By substituting 10 for n and 4 for r, the number of combinations can be calculated.
_nC_r=n!/r!(n-r)!
SubstituteII
n= 10, r= 4
_(10)C_4=10!/4!( 10- 4)!
▼
Evaluate right-hand side
SubTerm
Subtract term
_(10)C_4=10!/4!*6!
Write as a product
_(10)C_4=10*9*8*7*6!/4!*6!
CancelCommonFac
Cancel out common factors
_(10)C_4=10*9*8*7*6!/4! *6!
SimpQuot
Simplify quotient
_(10)C_4=10*9*8*7/4!
Write as a product
_(10)C_4=10*9*8*7/4*3*2*1
Multiply
Multiply
_(10)C_4=5040/24
CalcQuot
Calculate quotient
_(10)C_4=210
There are 210 ways in which Kriz can select 4 books to pack from their must-read list.
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Example
Finding Probabilities Using Combinations
Kriz has decided that they will select 5 of their books at random instead of 4. However, they would prefer to bring at least one book from the fantasy, mystery, and drama genres. What is the probability of them choosing these three genres if the selection pool consists of 10 books from 10 different genres? Write the answer in percentage form rounded to 1 decimal place.
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Hint
The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
Solution
The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
The order in which the books are selected is not important. Therefore, the number of possible outcomes can be found by calculating the combinations when taking 5 books out of 10. The number of combinations when selecting r items out of n is given by the following formula. _nC_r=n!/r!(n-r)! By substituting 10 for n and 5 for r, the number of possible outcomes can be calculated.
_nC_r=n!/r!(n-r)!
SubstituteII
n= 10, r= 5
_(10)C_5=10!/5!( 10- 5)!
▼
Evaluate right-hand side
SubTerm
Subtract term
_(10)C_5=10!/5!*5!
Write as a product
_(10)C_5=10*9*8*7*6*5!/5!*5!
CancelCommonFac
Cancel out common factors
_(10)C_5=10*9*8*7*6*5!/5!*5!
SimpQuot
Simplify quotient
_(10)C_5=10*9*8*7*6/5!
Write as a product
_(10)C_5=10*9*8*7*6/5*4*3*2*1
Multiply
Multiply
_(10)C_5=30240/120
CalcQuot
Calculate quotient
_(10)C_5=252
There are 252 ways of selecting 5 books out of 10. This is the number of possible outcomes. Possible Outcomes: 252 Since the order does not matter, there is only one way of selecting a fantasy, a mystery, and a drama book. The other 2 books need to be selected from the remaining 7. Therefore, the combinations of 2 out of 7 books will be calculated.
_nC_r=n!/r!(n-r)!
SubstituteII
n= 7, r= 2
_7C_2=7!/2!( 7- 2)!
▼
Evaluate right-hand side
SubTerm
Subtract term
_7C_2=7!/2!*5!
Write as a product
_7C_2=7*6*5!/2!*5!
CancelCommonFac
Cancel out common factors
_7C_2=7*6*5!/2!*5!
SimpQuot
Simplify quotient
_7C_2=7*6/2!
2!=2
_7C_2=7*6/2
Multiply
Multiply
_7C_2=42/2
CalcQuot
Calculate quotient
_7C_2=21
There is 1 way to select the first three books and 21 ways to select the other two books. By the Fundamental Counting Principle, the number of favorable outcomes is the product between 1 and 21.
Favorable Outcomes: 1*21= 21
There are 21 favorable outcomes and 252 possible outcomes. Let A be the event that 3 of the 5 books selected are a fantasy, a mystery, and a drama.
By substituting the number of favorable outcomes and the number of possible outcomes into the Probability Formula, P(A) can be found.
P(A)=Number of favorable outcomes/Number of possible outcomes
SubstituteII
Number of favorable outcomes= 21, Number of possible outcomes= 252
P(A)=21/252
ReduceFrac
a/b=.a /21./.b /21.
P(A)=1/12
FracToDiv
a/b=a÷ b
P(A)=0.083
RoundDec
Round to 3 decimal place(s)
P(A)≈ 0.083
WritePercent
Convert to percent
P(A)≈ 8.3 %
The probability of getting a fantasy, a mystery, and a drama in the five selected books is about 8.3 %.
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Example
The Birthday Problem
Magdalena teaches algebra to a group of 10 students. While making a list to track their attendance, she wonders whether at least 2 students have the same birthday. For simplicity, suppose that all years have exactly 365 days.
a What is the probability that at least two students have the same birthday? Approximate the answer to two decimal places.
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b If there were 30 students, what would be the probability that at least 2 students the same birthday? Approximate the answer to two decimal places.
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Hint
a Consider the Complement Rule of Probability.
b Substitute 30 for n and 2 for r in the expression found in Part A.
Solution
a Note that the phrase
Now, the numerator will be simplified.
The above expression represents the permutations of 365 taking n at a time. _(365)P_n=365!/(365-n)!
Therefore, P(A') is the quotient of _(365)P_n and 365^n.
P(A')=_(365)P_n/365^n
The Complement Rule of Probability states that P(A) is the difference between 1 and P(A'). By using the expression found for P(A'), an expression for P(A) can be found.
P(A)=1-P(A') ⇕ P(A)=1-_(365)P_n/365^n
Finally, by subtracting P(A') from 1, the probability of A can be calculated.
at leastmeans that any outcome with 2 or more students with the same birthday is a favorable outcome. Therefore, the following outcomes are all the possible favorable outcomes.
| Favorable Outcomes | ||
|---|---|---|
| 2 have the same birthday | 3 have the same birthday | 4 have the same birthday |
| 5 have the same birthday | 6 have the same birthday | 7 have the same birthday |
| 8 have the same birthday | 9 have the same birthday | 10 have the same birthday |
Suppose that instead of 10, there are n students. Let A be the event that at least 2 students out of n have their birthday on the same day. If n is greater than 365, then certainly at least 2 students would share their birthday. For the sake of the example, n is assumed to be less than or equal to 365. A=&At least 2students out of nhave &their birthday on the same day Finding the probability of every possible outcome — all nine outcomes in the table above — means finding nine different probabilities. The opposite of at least 2 students having their birthday on the same day is that there are no students with their birthday on the same day. This is the complement of A written as A'. A'=&No one of the nstudents &shares a birthday The Complement Rule of Probability can help to deal with this situation. To do so, a general expression will be found for no students from a group of n having the same birthday. Then, P(A) can be calculated.
General Expression For P(A') and P(A)
Suppose that the group consists of only 2 students. The first student can have their birthday on any day. The probability of the second student not having their birthday on the same day is the ratio of 364 to 365. Therefore, this expression is the probability of the two students not having their birthday on the same day. Probability of2 Students Not Having Their Birthday on the Same Day 364/365 Suppose that a third student is added to the group. The probability that this student has their birthday on a different day from the previous two is the ratio of 363 to 365. By the Multiplication Rule of Probability, the probability that no students out of these three have their birthdays on the same day can be found. Probability of3 Students Not Having Their Birthday on the Same Day 364/365*363/365 ⇔ 364*363/365^2 By following the same reasoning, the probability that n students do not share a birthday, P(A'), can be written. P(A')= 364/365*363/365*...365-n+1/365_(n-1 times) [2.5em] ⇕ [1em] P(A')= 364*363*...*(365-n+1)/365^(n-1) The obtained expression will now be simplified.
P(A')=364*363*...*(365-n+1)/365^(n-1)
▼
Simplify right-hand side
IdPropMult
Identity Property of Multiplication
P(A')=1* 364*363*...*(365-n+1)/365^(n-1)
OneToFrac
Rewrite 1 as 365/365
P(A')=365/365* 364*363*...*(365-n+1)/365^(n-1)
MultFrac
Multiply fractions
P(A')=365* 364*363*...*(365-n+1)/365* 365^(n-1)
MultBasePow
a*a^m=a^(1+m)
P(A')=365* 364*363*...*(365-n+1)/365^n
365* 364*363*...*(365-n+1)
▼
Simplify
IdPropMult
Identity Property of Multiplication
365* 364*363*...*(365-n+1) * 1
OneToFrac
Rewrite 1 as (365-n)!/(365-n)!
365* 364*363*...*(365-n+1) * (365-n)!/(365-n)!
MoveLeftFacToNum
a*b/c= a* b/c
365* 364*363*...*(365-n+1) * (365-n)!/(365-n)!
Substitute
365* 364*363*...*(365-n+1) * (365-n)!= 365!
365!/(365-n)!
Calculating P(A) if n=10
In this case, A is the event that at least 2 students out of 10 have their birthday on the same day.
| A | A' |
|---|---|
| At leat 2 students out of 10 have their birthday on the same day. | No one of the 10 students shares a birthday. |
To apply the formula, P(A') will be calculated first.
P(A')=_(365)P_n/365^n
Substitute
n= 10
P(A')=_(365)P_(10)/365^(10)
▼
Evaluate right-hand side
Substitute
_(365)P_(10)= 365!/(365-10)!
P(A')=365!(365-10)!/365^(10)
SubTerm
Subtract term
P(A')=365!355!/365^(10)
Write as a product
P(A')=365*364*...*356*355!355!/365^(10)
CancelCommonFac
Cancel out common factors
P(A')=365*364*...*356*355!355!/365^(10)
SimpQuot
Simplify quotient
P(A')=365*364*...*3561/365^(10)
DivByOne
a/1=a
P(A')=365*364*...*356/365^(10)
UseCalc
Use a calculator
P(A')=0.883051...
RoundDec
Round to 2 decimal place(s)
P(A')≈ 0.88
b It was found that if A is the event that at least 2 out of n students have their birthday on the same day, P(A) is given by the following equation.
The probability that at least 2 students out of 30 have their birthday on the same day is about 0.71. What is more, and extremely curious, only 57 people are required to bring the probability that at least two people have the same birthday up to 0.99.
P(A)=1-_(365)P_n/365^n By substituting 30 for n, P(A) can be found for a group of 30 students.
P(A)=1-_(365)P_n/365^n
Substitute
n= 30
P(A)=1-_(365)P_(30)/365^(30)
▼
Evaluate right-hand side
Substitute
_(365)P_(30)= 365!/(365-30)!
P(A)=1-365!(365-30)!/365^(30)
SubTerm
Subtract term
P(A)=1-365!335!/365^(30)
Write as a product
P(A)=1-365*364...*336*335!335!/365^(30)
CancelCommonFac
Cancel out common factors
P(A)=1-365*364...*336*335!335!/365^(30)
SimpQuot
Simplify quotient
P(A)=1-365*364...*3361/365^(30)
DivByOne
a/1=a
P(A)=1-365*364...*336/365^(30)
UseCalc
Use a calculator
P(A)=1-0.293683...
SubTerm
Subtract term
P(A)=0.706317...
RoundDec
Round to 2 decimal place(s)
P(A)≈ 0.71
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Closure
Creating Arrangements
Permutations and combinations can be used in many situations. Understanding these mathematical concepts can help solve many intricate problems. With this in mind, reconsider the problem in which Vincenzo wants to create arrangements by using the following letters.
How many different arrangements with 3 vowels and 4 consonants can be created?
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Hint
Begin by calculating the number of ways of selecting 3 vowels and 4 consonants. The order of the arrangements is essential.
Solution
Because the arrangements consist of 3 vowels and 4 consonants, they have 7 letters. An example arrangement is shown.
Note that 3 vowels must be selected out of 5, which means that the number of possible combinations must be calculated. To do so, the combination formula can be used. The number of combinations of n objects taken r at a time is given by the following formula. _nC_r=n!/r!(n-r)! Therefore, the number of possible combinations can be calculated by substituting 5 for n and 3 for r into the formula.
_nC_r=n!/r!(n-r)!
SubstituteII
n= 5, r= 3
_5C_3=5!/3!( 5- 3)!
▼
Evaluate right-hand side
SubTerm
Subtract term
_5C_3=5!/3!*2!
Write as a product
_5C_3=5*4*3!/3!*2!
CancelCommonFac
Cancel out common factors
_5C_3=5*4*3!/3!*2!
SimpQuot
Simplify quotient
_5C_3=5*4/2!
Multiply
Multiply
_5C_3=20/2!
2!=2
_5C_3=20/2
CalcQuot
Calculate quotient
_5C_3=10
In a similar way, the number of combinations when taking 4 out of 6 constants can be found.
_nC_r=n!/r!(n-r)!
SubstituteII
n= 6, r= 4
_6C_4=6!/4!( 6- 4)!
▼
Evaluate right-hand side
SubTerm
Subtract term
_6C_4=6!/4!*2!
Write as a product
_6C_4=6*5*4!/4!*2!
CancelCommonFac
Cancel out common factors
_6C_4=6*5*4!/4!*2!
SimpQuot
Simplify quotient
_6C_4=6*5/2!
Multiply
Multiply
_6C_4=30/2!
2!=2
_6C_4=30/2
CalcQuot
Calculate quotient
_6C_4=15
There are 10 ways of selecting three vowels and 15 ways of selecting four consonants. By the Fundamental Counting Principle, the number of ways of selecting three vowels and four consonants is given by the product of 10 and 15. 10*15= 150 Now, keep in mind that a different arrangement of each possible combination is different. This means that the order is important, and the number of possible permutations of the seven letters must to be calculated. Recall that the factorial of n gives the number of permutations of n out of n. _7P_7=7! ⇔ _7P_7= 5040 Finally the Fundamental Counting Principle will be used one more time. For each of the 150 ways of selecting three vowels and four consonants, there are 5040 permutations. Possible Arrangements 150* 5040=756 000
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Permutations and Combinations in Probability
Exercise 3.1
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Assuming that Tiffaniqua gets her aisle seat, there are five seats left. Note that there are two aisle seats, which means that she has two choices of where to sit.
The position in which the friends will sit is essential, so this exercise involves permutations. Additionally, Tiffaniqua has 2 options where to sit. After this, there are 5 people to arrange in the 5 remaining seats. Recall that the permutation of n items out of n is given by the factorial of n. _nP_n=n! Therefore, we need to calculate the factorial of 5 to find the number of ways we can arrange the other five people. _5P_5=5! ⇔ _5P_5= 120 There are 2 ways in which Tiffaniqua can sit and 120 ways in which Tiffaniqua's friends can sit. From the Fundamental Counting Principle, the product of these values gives the number of ways the group can sit down if Tiffaniqua has an aisle seat. 2* 120=240 ways
The probability of an event A is given by the ratio of the favorable outcomes to the number of possible outcomes. P(A)=Number of favorable outcomes/Number of possible outcomes Note that the number of different arrangements of the 6 friends corresponds to the number of possible outcomes. Because we are want to arrange 6 out of 6 people, the number of possible arrangements is given by the factorial of 6. Number of favorable outcomes=6! ⇕ Number of favorable outcomes= 720 To find the number of favorable outcomes, we need to determine the number of arrangements where two boys occupy the seats on the left and four girls occupy the seats on the right. These permutations can be expressed as _2P_2 and _4P_4. Let's use our calculator to find these permutations.
We can see that there are 2 ways the boys can occupy the seats on the left. Similarly, there are 24 ways that the girls can occupy the seats on the right. By multiplying these values, we will find the number of favorable outcomes. 2*24= 48 Now, let's call A the event that the four girls will seat on the right and the two boys on the left. We can now evaluate the formula to find the probability of A.
Therefore, the probability of ending up with the four girls on the right and the two boys on the left is 115.
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We first need to select the books from each set according to the number required and then arrange the books on the shelf. The word select
means combinations. The number of combinations can be found using the combination formula.
_nC_r=n!/r!(n-r)!
To find the number of combinations when 2 fantasy books out of 5 are selected, we need to evaluate the formula when r is 2 and n is 5. Let's do it!
In a similar way, we can find the number of combinations for the biography and history books. Let's do this on a table.
| n | r | _nC_r=n!/r!(n-r)! | Simplification |
|---|---|---|---|
| 5 | 2 | _5C_2=5!/2!( 5- 2)! | 10 |
| 6 | 3 | _6C_3=6!/3!( 6- 3)! | 20 |
| 4 | 2 | _4C_2=4!/2!( 4- 2)! | 6 |
We have found that there are 10 possible ways to choose a combination of 2 fantasy books (F), 20 combinations of 3 biography books (B) and 6 combinations of 2 history books (H). This gives a total of 7 books selected. 2 F + 3B+ 2H= 7Books To find the number of combinations of these 7 books, we can use the Fundamental Counting Principle. This means we need to multiply the ways in which each type of book can be selected. 10*20*6= 1200 combinations We have found that the seven books can be combined in 1200 ways. However, a single combination can have multiple ways of arranging the books on the shelf. This means we need to find the number of permutations of 7 books out of 7. The number of permutations of n objects out of n is given by the factorial of n. _nP_n=n! Therefore, we need to find the factorial of 7. _7P_7=7! ⇔ _7P_7= 5040 However, this represents only the number of arrangements of 1 of the 1200 possible combinations. We can use the Fundamental Counting Principle again to find the total number of arrangements. This means we need to multiply 1200 by 5040 1200* 5040 = 6 048 000arrangements Therefore, Diego can arrange two fantasy books, three biographies, and two history books on a shelf in 6 048 000 ways.
We are asked to find the probability that the 2 fantasy books (F) are not arranged together on the shelf. The probability of an event A is given by the ratio of the number of favorable outcomes to the number of possible outcomes. P(A)= Number of favorable outcomes/Number of possible outcomes Note that the favorable outcomes are all the outcomes in which the books are not in consecutive places. For example, if the first fantasy book is in the first place, the second can be in any remaining places except in the second place. F F F F ... F F This represents the outcomes when one of the fantasy books is in the first place. However, it can also be in second place, third place, and so on. This makes it difficult to analyze and count all the favorable outcomes. Still, we have a simpler way of solving this problem. This can be done by using the complement rule of probability. P(A)=1-P(A') The complement rule states that the probability of an event A can be found by subtracting the probability of its complement A' from 1. Now, let's note that the opposite of having two fantasy books in nonconsecutive placements is that the books are together. A:&Fantasy books are not together [0.5em] A':&Fantasy books are together In this case, for A', the favorable outcomes are the following. F F F F F F F F F F F F Note that we can count the two books as one item to ensure they are arranged together. In doing so, we now have 6 items to arrange. We can find the number of arrangements by calculating the number of permutations when arranging 6 out of 6 items. _6P_6=6! ⇔ _6P_6= 720 However, we still need to consider that the two books can be arranged in 2 ways. Therefore, we need to multiply the number of permutations by the number of ways the two books can be arranged. 2* 720= 1440 Furthermore, 1440 only represents the permutations of one of the many possible combinations of the original 7 books in which the two fantasy books are arranged together. Previously, we have found that all the possible combinations are given by the product of the ways in which the types of books can be selected. 10*20*6= 1200combinations Therefore, by the Fundamental Counting Principle, multiplying the number of permutations in which the fantasy books are together by the total number of possible selections will give us the favorable outcomes for A'. Number of Favorable Outcomes &= 1440* 1200 &= 1 728 000 Moreover, the number of possible outcomes is given by the ways that two fantasy books, three biographies, and two history books can be arranged on the shelf. We previously found this is to be 6 048,000. We can now apply the probability formula to find P(A').
Finally, by substituting this value into the complement rule, the probability of the fantasy books not being arranged together can be found.
Therefore, the probability that the fantasy books are not together in the shelf is 57.
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Permutations and Combinations in Probability
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