Permutations - Algebra 2
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An ice cream vendor sells five different flavors of ice cream.
In how many ways can you choose three scoops of different ice cream flavors if order matters?
An ice cream vendor sells five different flavors of ice cream.
In how many ways can you choose three scoops of different ice cream flavors if order matters?
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There are five ways to choose the first scoop, then four ways to choose the second scoop, and finally three ways to choose the third scoop:
5 * 4 * 3 = 60
There are five ways to choose the first scoop, then four ways to choose the second scoop, and finally three ways to choose the third scoop:
5 * 4 * 3 = 60
There are 5 men and 4 women competing for an executive body consisting of :
- President
- Vice President
- Secretary
- Treasurer
It is required that 2 women and 2 men must be selected
How many ways the executive body can be formed?
There are 5 men and 4 women competing for an executive body consisting of :
- President
- Vice President
- Secretary
- Treasurer
It is required that 2 women and 2 men must be selected
How many ways the executive body can be formed?
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2 men can be selected:
2 women can be selected out of 4 women:
Finally, after the selection process, these men and women can fill the executive body in 
ways.
This gives us a total of 
2 men can be selected:
2 women can be selected out of 4 women:
Finally, after the selection process, these men and women can fill the executive body in
This gives us a total of
Find the Computing Permutation.
Find the Computing Permutation.
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There are 
runners in a race. How many different arrangements are there for 
, 
, and 
place?
There are 
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This is a permutation of 10 objects (runners) taken 3 at a time, with no replacements.
Another way to look at this would be there are 10 runners competing for 1st place, 9 runners competing for 2nd place, and 8 runners competing for 3rd place.
This is a permutation of 10 objects (runners) taken 3 at a time, with no replacements.
Another way to look at this would be there are 10 runners competing for 1st place, 9 runners competing for 2nd place, and 8 runners competing for 3rd place.
How many ways can a three committee board select the president, vice president and treasurer from a group of 15 people?
How many ways can a three committee board select the president, vice president and treasurer from a group of 15 people?
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In this problem, order is important because once someone is chosen as a position they can not be chosen again, and once a position is filled, no one else can fill that in mind.
The presidential spot has a possibility of 15 choices, then 14 choices for vice president and 13 for the treasurer.
So:
In this problem, order is important because once someone is chosen as a position they can not be chosen again, and once a position is filled, no one else can fill that in mind.
The presidential spot has a possibility of 15 choices, then 14 choices for vice president and 13 for the treasurer.
So:
How many ways can you re-arrange the letters of the word JUBILEE?
How many ways can you re-arrange the letters of the word JUBILEE?
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There are 7 letters in the word jubilee, so initially we can calculate that there are 
ways to re-arrange those letters. However, The letter e appears twice, so we're double counting. Divide by 2 factorial (2) to get 
.
There are 7 letters in the word jubilee, so initially we can calculate that there are
How many ways can you re-arrange the letters of the word BANANA?
How many ways can you re-arrange the letters of the word BANANA?
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At first, it makes sense that there are 
ways to re-arrange these letters. However, the letter A appears 3 times and the letter N appears twice, so divide first by 3 factorial and then 2 factorial:
At first, it makes sense that there are
In a class of 24 students, how many distinct groups of 4 can be formed?
In a class of 24 students, how many distinct groups of 4 can be formed?
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To solve, evaluate 
To solve, evaluate
7 students try out for the roles of Starsky and Hutch in a new school production. How many different ways can these roles be cast?
7 students try out for the roles of Starsky and Hutch in a new school production. How many different ways can these roles be cast?
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There are 7 potential actors and 2 different roles to fill. This would be calculated as 
divided by 
, or 42
There are 7 potential actors and 2 different roles to fill. This would be calculated as
13 rubber ducks are competing in a race. How many different arrangements of first, second, and third place are possible?
13 rubber ducks are competing in a race. How many different arrangements of first, second, and third place are possible?
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There are 3 winners out of the total set of 13. That means we're calculating 
There are 3 winners out of the total set of 13. That means we're calculating
How many different 4 letter words can be made out of the letters A, B, C, D, and E?
How many different 4 letter words can be made out of the letters A, B, C, D, and E?
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Since order matters, use the permutation formula:
There are 5 letters to choose from (n), and you pick 4 of them (r).
So there are 120 possible words that can be formed.
Since order matters, use the permutation formula:
There are 5 letters to choose from (n), and you pick 4 of them (r).
So there are 120 possible words that can be formed.
The soccer team awards gold, silver, and bronze trophies for the top three goal scorers over the season. If the soccer team has 11 players, how many different ways could the gold, silver, and bronze trophies be awarded?
The soccer team awards gold, silver, and bronze trophies for the top three goal scorers over the season. If the soccer team has 11 players, how many different ways could the gold, silver, and bronze trophies be awarded?
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Because the order is important, this is a permutation. There are 11 players to fill 3 spots, so 
.
Because the order is important, this is a permutation. There are 11 players to fill 3 spots, so
How many possible ways can the letters A and B be assigned to nine people?
How many possible ways can the letters A and B be assigned to nine people?
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Write the formula for permutation.
Evaluate 
.
Cancel all the common terms in the numerator and denominator.
The answer is: 
Write the formula for permutation.
Evaluate
Cancel all the common terms in the numerator and denominator.
The answer is:
Evaluate: 
Evaluate:
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Write the permutation formula.
Substitute the values in the given problem.
Cancel out the terms in the numerator and denominator.
The remaining terms in the numerator are:
The answer: 
Write the permutation formula.
Substitute the values in the given problem.
Cancel out the terms in the numerator and denominator.
The remaining terms in the numerator are:
The answer:
Evaluate: 
Evaluate:
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Write the permutation formula.
Substitute the values of the variables.
Simplify the factorials.
Cancel the common terms.
The answer is: 
Write the permutation formula.
Substitute the values of the variables.
Simplify the factorials.
Cancel the common terms.
The answer is:
Determine the value of: 
Determine the value of:
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Write the permutation formula.
Substitute the values of 
into the formula.
Expand the terms.
Cancel out the terms in the denominator and denominator.
The remaining terms are: 
Multiply these numbers.
The answer is: 
Write the permutation formula.
Substitute the values of
Expand the terms.
Cancel out the terms in the denominator and denominator.
The remaining terms are:
Multiply these numbers.
The answer is:
Solve the permutation: 
Solve the permutation:
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Write the formula for permutation.
Substitute the values and expand the factorials.
Reduce the terms.
The answer is: 
Write the formula for permutation.
Substitute the values and expand the factorials.
Reduce the terms.
The answer is:
Suppose a locksmith designs a three digit lock with numbers ranging only from zero to five. What is the number of permutations possible?
Suppose a locksmith designs a three digit lock with numbers ranging only from zero to five. What is the number of permutations possible?
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The formula to determine the permutation is:
Notice that for each number, there are 6 different choices since we have numbers ranging from zero to five, not five.
For a three digit lock, 
.
Substitute the terms into the formula.
The answer is: 
The formula to determine the permutation is:
Notice that for each number, there are 6 different choices since we have numbers ranging from zero to five, not five.
For a three digit lock,
Substitute the terms into the formula.
The answer is:
What is the probability of flipping exactly one heads and exactly one tails in two coin tosses?
What is the probability of flipping exactly one heads and exactly one tails in two coin tosses?
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Each of the two coin flips represents one event. The probability of obtaining exactly one heads and exactly one tails can be modeled as follows:
Event 1 * Event 2
The key to understanding this problem is to recognize that either heads followed by tails, or tails followed by heads, would satisfy the specific overall outcome asked for in the problem. Since Event 1 will produce either a heads or a tails, we have a 100% chance of obtaining an outcome from Event 1 that will satisfy one of the two requirements of the specific overall outcome (one heads and one tails). Event 2 will then have a 50% chance of producing an outcome that is the opposite, rather than the same, as the outcome of Event 1. We can therefore calculate the probability of our specific overall outcome as follows:
100% * 50%
1 * 0.5
0.5 = 50%
Therefore, the probability of obtaining one heads and one tails from two coin tosses is 50%.
Each of the two coin flips represents one event. The probability of obtaining exactly one heads and exactly one tails can be modeled as follows:
Event 1 * Event 2
The key to understanding this problem is to recognize that either heads followed by tails, or tails followed by heads, would satisfy the specific overall outcome asked for in the problem. Since Event 1 will produce either a heads or a tails, we have a 100% chance of obtaining an outcome from Event 1 that will satisfy one of the two requirements of the specific overall outcome (one heads and one tails). Event 2 will then have a 50% chance of producing an outcome that is the opposite, rather than the same, as the outcome of Event 1. We can therefore calculate the probability of our specific overall outcome as follows:
100% * 50%
1 * 0.5
0.5 = 50%
Therefore, the probability of obtaining one heads and one tails from two coin tosses is 50%.
An ice cream vendor sells five different flavors of ice cream.
In how many ways can you choose three scoops of different ice cream flavors if order matters?
An ice cream vendor sells five different flavors of ice cream.
In how many ways can you choose three scoops of different ice cream flavors if order matters?
Tap to see back →
There are five ways to choose the first scoop, then four ways to choose the second scoop, and finally three ways to choose the third scoop:
5 * 4 * 3 = 60
There are five ways to choose the first scoop, then four ways to choose the second scoop, and finally three ways to choose the third scoop:
5 * 4 * 3 = 60