Combinations with Repetition

Definition

Combinations with repetition is a concept in probability theory that deals with the number of ways to select a certain number of items from a set, where the order of selection does not matter and the same item can be selected more than once.

5 Must Know Facts For Your Next Test

1. Combinations with repetition are used to calculate the number of possible outcomes when selecting items from a set where the same item can be selected more than once.
2. The formula for calculating combinations with repetition is $n^r$, where $n$ is the number of items in the set and $r$ is the number of items being selected.
3. Combinations with repetition are often used in problems involving the selection of items from a set with replacement, such as drawing cards from a deck or rolling dice.
4. The order of selection does not matter in combinations with repetition, so the number of possible outcomes is not affected by the order in which the items are selected.
5. Combinations with repetition can be used to model a wide range of real-world situations, such as the number of ways to choose a combination lock, the number of ways to distribute a set of items among a group of people, or the number of ways to select a team from a larger group.

Review Questions

• Explain the difference between combinations with repetition and permutations.
  • The key difference between combinations with repetition and permutations is the importance of order. In permutations, the order of selection matters, so the number of possible outcomes is affected by the order in which items are chosen. In combinations with repetition, the order of selection does not matter, and the same item can be selected more than once. The formula for combinations with repetition is $n^r$, where $n$ is the number of items in the set and $r$ is the number of items being selected, while the formula for permutations is $n!/(n-r)!$, where $n$ is the number of items in the set and $r$ is the number of items being selected.
• Describe a real-world scenario where combinations with repetition would be applicable.
  • One real-world scenario where combinations with repetition would be applicable is the problem of choosing a combination lock. Suppose a combination lock has 4 dials, each with 10 possible numbers (0-9). To calculate the number of possible combinations, you would use the formula for combinations with repetition, $n^r$, where $n$ is the number of possible numbers on each dial (10) and $r$ is the number of dials (4). This would give you $10^4 = 10,000$ possible combinations. This demonstrates how combinations with repetition can be used to model real-world situations involving the selection of items from a set where the same item can be chosen more than once.
• How can the concept of combinations with repetition be used to model the distribution of items among a group of people?
  • The concept of combinations with repetition can be used to model the number of ways to distribute a set of items among a group of people, where the order of distribution does not matter and the same person can receive multiple items. For example, if there are 10 items to be distributed among 5 people, the number of possible distributions can be calculated using the formula for combinations with repetition, $n^r$, where $n$ is the number of items (10) and $r$ is the number of people (5). This would give you $10^5 = 100,000$ possible distributions. This demonstrates how the principles of combinations with repetition can be applied to real-world situations involving the distribution of resources or items among a group.

Related terms

Permutations:

Permutations are the number of ways to arrange a set of items in a specific order, where the order of selection matters.

Combinations:

Combinations are the number of ways to select a certain number of items from a set, where the order of selection does not matter.

Probability:

Probability is the measure of the likelihood of an event occurring, expressed as a number between 0 and 1.