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Here are a few recommended readings before getting started with this lesson.
Vincenzo is playing with the following letters.
He wants to create as many different arrangements as possible using 7 of the letters without repeating any letters in each arrangement. He also decides that the arrangements must consist of 3 vowels and 4 consonants. How many different arrangements can Vincenzo create?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.
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.Listing all the permutations may be a difficult task when many objects are being arranged. In these cases, the Permutation Formula can be used instead.
The number of permutations of n different objects arranged r at a time — denoted as nPr — is given by the following formula.
nPr=(n−r)!n!,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.
nPn=n!
An alternative notation for nPr is P(n,r).
The 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) |
Write as a product
Write as a factorial
LHS/(n−r)!=RHS/(n−r)!
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 |
n=10
Write as a product
Multiply
n=10, r=3
Subtract term
Write as a product
Cancel out common factors
Simplify quotient
1a=a
Multiply
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 |
n=8, r=3
Subtract term
Write as a product
Cancel out common factors
Simplify quotient
1a=a
Multiply
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 |
Substitute values
ba=b/120a/120
ba=a÷b
Round to 3 decimal place(s)
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.
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.
Because the order of the items does not matter, two combinations are different from each other if they do not have the same objects. The number of combinations can be found by listing every possible combination. However, this method is not helpful when considering a large number of objects. The Combination Formula should be used instead.The number of combinations of n different objects taken r at a time — denoted as nCr — is given by the following formula.
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.
An alternative notation for cCr is C(n,r).
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 |
The order in which the books are selected is not crucial.
n=10, r=4
Subtract term
Write as a product
Cancel out common factors
Simplify quotient
Write as a product
Multiply
Calculate quotient
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.
The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
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.n=10, r=5
Subtract term
Write as a product
Cancel out common factors
Simplify quotient
Write as a product
Multiply
Calculate quotient
n=7, r=2
Subtract term
Write as a product
Cancel out common factors
Simplify quotient
2!=2
Multiply
Calculate quotient
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.
Number of favorable outcomes=21, Number of possible outcomes=252
ba=b/21a/21
ba=a÷b
Round to 3 decimal place(s)
Convert to percent
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.
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 |
Identity Property of Multiplication
Rewrite 1 as 365365
Multiply fractions
a⋅am=a1+m
Identity Property of Multiplication
Rewrite 1 as (365−n)!(365−n)!
a⋅cb=ca⋅b
365⋅364⋅363⋅…⋅(365−n+1)⋅(365−n)!=365!