Podcast
Questions and Answers
In a scenario where you are selecting a team of 3 members from a group of 10, but the order in which you select them determines their roles (leader, strategist, and communicator), which formula should be used to calculate the number of possible teams?
In a scenario where you are selecting a team of 3 members from a group of 10, but the order in which you select them determines their roles (leader, strategist, and communicator), which formula should be used to calculate the number of possible teams?
- Factorial formula because all members are being arranged.
- A simple multiplication rule because each member is being selected independently.
- Combinations formula because order does not matter.
- Permutations formula because order matters. (correct)
What distinguishes a permutation problem from a combination problem?
What distinguishes a permutation problem from a combination problem?
- Combinations allow repetition, while permutations do not.
- Permutations consider the order of items, while combinations do not. (correct)
- Permutations involve selecting items, while combinations involve arranging items.
- Combinations consider the order of items, while permutations do not.
In how many ways can the letters of the word 'SUCCESS' be arranged such that the arrangement starts and ends with the letter 'S'?
In how many ways can the letters of the word 'SUCCESS' be arranged such that the arrangement starts and ends with the letter 'S'?
- 60 (correct)
- 120
- 360
- 720
A restaurant offers a 'build-your-own-pizza' special with 10 different toppings. If a customer can choose any combination of toppings, including no toppings at all, how many different pizzas are possible?
A restaurant offers a 'build-your-own-pizza' special with 10 different toppings. If a customer can choose any combination of toppings, including no toppings at all, how many different pizzas are possible?
In what scenario would using a permutation calculation be most appropriate?
In what scenario would using a permutation calculation be most appropriate?
A website requires users to create a password that is exactly 8 characters long. The password must contain at least one digit, one uppercase letter, and one lowercase letter. Repeated characters are allowed. Which of the following methods would be most appropriate to calculate the number of possible passwords?
A website requires users to create a password that is exactly 8 characters long. The password must contain at least one digit, one uppercase letter, and one lowercase letter. Repeated characters are allowed. Which of the following methods would be most appropriate to calculate the number of possible passwords?
A software company needs to create a password system. If the password must be 8 characters long and can consist of any combination of upper and lower case letters (26 each) and numbers (10), how many different passwords can they create if repetition is allowed?
A software company needs to create a password system. If the password must be 8 characters long and can consist of any combination of upper and lower case letters (26 each) and numbers (10), how many different passwords can they create if repetition is allowed?
A researcher wants to analyze the effects of different drug combinations on patients. If she has 10 different drugs and wants to test every possible combination of 3 drugs at a time on different patients, how many patient groups will she need?
A researcher wants to analyze the effects of different drug combinations on patients. If she has 10 different drugs and wants to test every possible combination of 3 drugs at a time on different patients, how many patient groups will she need?
A bakery offers 15 different kinds of donuts. If you want to select a box of 4 donuts, what is the number of different selections you can make, assuming that the order in which you place the donuts in the box is irrelevant, and you can choose the same kind of donut multiple times?
A bakery offers 15 different kinds of donuts. If you want to select a box of 4 donuts, what is the number of different selections you can make, assuming that the order in which you place the donuts in the box is irrelevant, and you can choose the same kind of donut multiple times?
In how many ways can the letters of the word 'ARRANGE' be arranged such that the two 'R's are always together?
In how many ways can the letters of the word 'ARRANGE' be arranged such that the two 'R's are always together?
Flashcards
Permutation
Permutation
An arrangement of items where the order matters.
Combination
Combination
A selection of items where the order does not matter.
Permutation keywords
Permutation keywords
Arrangements, lists, sequences
Combination keywords
Combination keywords
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Factorial
Factorial
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Permutations Formula
Permutations Formula
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Combinations Formula
Combinations Formula
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Permutations with Repetition
Permutations with Repetition
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Study Notes
- Permutations and combinations are arrangements of items like people, numbers, or objects.
- Permutations are ordered arrangements.
- Combinations are unordered arrangements.
Key Differences
- Permutations consider order; combinations do not.
- Arrangements, lists, and sequences indicate permutation problems.
- Groups, collections, and sets suggest combination problems.
- Committees exemplify combination problems because order is irrelevant
Permutation and Combination Formulas
- There exist five formulas used to calculate permutations/combinations.
- Selecting the correct formula requires considering how choices are made.
- Key questions to ask: "Does order matter?" and "Is repetition allowed?"
- Factorial notation is used in many formulas, where '!' denotes the product of all integers less than or equal to that number.
Permutations Formulas
- Arranging objects chosen from objects, multiply by itself times ( calculate ).
- Order matters, repetition is allowed: arrangements.
- Order matters, repetition is not allowed: use the Permutations formula:
- Arranging all objects (order matters, no repetition): calculate
- Permutations formula: simplifies to a product of numbers.
Combinations Formulas
- In combinations, order is irrelevant, so the number of combinations is generally less than permutations.
- "ABC" and "CBA" are the same in combinations but different in permutations.
- Order does not matter, Repetition is allowed: There are : arrangements.
- Order does not matter, Repetition is not allowed: use the Combinations formula:
Solving Problems
- Eight racers, find the number of ways to assign gold, silver, bronze medals - use permutations.
- Eight racers in a qualifying heat; selection of the top 3 matters, but not their order - use combinations.
Solving Permutation Problems with Repetition
- Determine the number of 4-digit passcodes possible (order matters, repetition allowed) - Use where n = 10.
- Determining the number of possible passcode numbers where order doesn't matter and repetition is allowed is rare.
Example Problems
Example 1
- Find the number of ways to arrange the letters in the word "DOUBLE".
- Order matters, repetition is not allowed.
- Solve by calculating 6! or 6 x 5 x 4 x 3 x 2 x 1 = 720.
Example 2
- Lucy has 5 books and wants to choose 2 for a trip.
- Combination problem where order doesn't matter, and repetition isn't allowed.
- Solve using the combinations formula 5! / (2! * (5-2)!) = 10.
Example 3
- Select and arrange 5 out of 8 quiz questions.
- Permutation problem where order matters, and repetition isn't allowed.
- Solve using the permutations formula: 8! / (8-5)!= 6,720
Example 4
- License plate with 4 characters, each a letter or a digit.
- Permutation problem where order matters, and repetition is allowed.
- Solve using formula , where = 36 (26 letters + 10 digits),
- Therefore it is expressed as = 1,679,616.
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Description
Understand the difference between permutations and combinations. Permutations are ordered arrangements, while combinations are unordered. Learn to identify the correct formula based on whether order and repetition matter.
