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Questions and Answers
What is the key characteristic of combinations?
What is the key characteristic of combinations?
- All items must be arranged in a circle.
- Selection involves distinct items only.
- Order does not matter in selection. (correct)
- Order matters in selection.
How many ways can 3 students be selected from a group of 10?
How many ways can 3 students be selected from a group of 10?
- 210 ways
- 120 ways (correct)
- 30 ways
- 100 ways
In how many ways can 5 students be arranged from a group of 10?
In how many ways can 5 students be arranged from a group of 10?
- 1000 ways
- 120 ways
- 30240 ways (correct)
- 50 ways
Which situation illustrates a combination?
Which situation illustrates a combination?
What is the primary difference between combinations and permutations?
What is the primary difference between combinations and permutations?
Which of the following applications is best suited for permutations?
Which of the following applications is best suited for permutations?
How many arrangements can be made for 4 distinct trophies on a shelf?
How many arrangements can be made for 4 distinct trophies on a shelf?
What is the number of ways to select a committee of 4 members from a group of 15 people?
What is the number of ways to select a committee of 4 members from a group of 15 people?
Which of the following scenarios requires the use of permutations?
Which of the following scenarios requires the use of permutations?
How many distinguishable permutations are there for the letters in the word 'BALLOON'?
How many distinguishable permutations are there for the letters in the word 'BALLOON'?
What would be the result of calculating 5P3?
What would be the result of calculating 5P3?
Which of the following represents the number of combinations of choosing 3 objects from a set of 7?
Which of the following represents the number of combinations of choosing 3 objects from a set of 7?
If a password is created using 4 distinct letters from the alphabet and the order matters, how many different permutations can be formed?
If a password is created using 4 distinct letters from the alphabet and the order matters, how many different permutations can be formed?
In how many ways can a selection of 5 players be made from a team of 15 without regard to their positions?
In how many ways can a selection of 5 players be made from a team of 15 without regard to their positions?
When arranging 4 different colors in a row, how many different permutations are possible?
When arranging 4 different colors in a row, how many different permutations are possible?
Flashcards
Combinations
Combinations
Number of ways to choose items without order.
Permutations
Permutations
Number of ways to arrange items in order.
Order in Combinations
Order in Combinations
Order of selection doesn't matter.
Order in Permutations
Order in Permutations
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Combination Example
Combination Example
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Permutation Example
Permutation Example
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Fundamental Counting Principle
Fundamental Counting Principle
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Distinguishable Permutations
Distinguishable Permutations
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Study Notes
Combination
- Combinations calculate the number of ways to choose a subset of items from a set, where the order of selection does not matter.
- The formula for combinations is given by: nCr, where 'n' is the total number of items and 'r' is the number of items to be chosen.
- Example: Selecting 3 students from a group of 10 students for a project. The order in which the students are selected doesn't influence which group is formed.
- Key characteristic: Order does not matter. A selection of A, B, and C is the same as C, B, and A.
Permutation
- Permutations calculate the number of ways to arrange a set of items in a specific order.
- The formula for permutations is given by: nPr, where 'n' is the total number of items and 'r' is the number of items to be arranged.
- Example: Assigning different tasks to 3 people from a group of 10 people, where each task is handled by a different person. The order of assignment matters.
- Key characteristic: Order matters. A, B, and C is different from C, B, and A.
Key Differences between Combinations and Permutations
- Order: Combinations do not consider the order of selection, whereas permutations do.
- Selection vs Arrangement: Combinations focus on selecting a subset, whereas permutations focus on arranging the chosen items in a specific order.
- Formula: The formula for combinations is different from the formula for permutations.
Applications
- Combinations: Determining the number of possible lottery tickets, selecting a committee from a group of people, evaluating game outcomes.
- Permutations: Scheduling events, arranging books on a shelf, determining different seating arrangements, coding.
Examples
- Combination Example: How many ways can 3 flavors of ice cream be chosen from 10 flavors if order doesn't matter?
- n = 10, r = 3
- 10C3 = 120 ways
- Permutation Example: In how many ways can 5 out of 10 students be arranged in a row?
- n = 10, r = 5
- 10P5 = 30240 ways
Additional Considerations
- Distinct Objects: The above formulas assume all objects to be distinct.
- Repetition: When items can be repeated, the formulas change. The study of permutations and combinations with repetitions is a separate topic.
- Circular Permutations: In some cases, the arrangement is in a circle, which affects the calculation. These methods have different formulas.
- Important Note: Factorials (like 5!, often denoted as 5 factorial) are the product of all positive integers from 1 to a given number (e.g., 5! = 5 x 4 x 3 x 2 x 1 = 120). Knowledge of factorials is essential for applying the formulas.
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