Combinations with repetition

Definition

Combinations with repetition refers to the selection of items from a set where the same item can be chosen more than once, and the order of selection does not matter. This concept is essential in combinatorial mathematics, particularly when dealing with problems that involve choosing subsets from larger sets while allowing for duplicates. It expands on the idea of regular combinations by accounting for scenarios where repetitions are allowed, thus increasing the number of possible selections.

5 Must Know Facts For Your Next Test

1. The formula for calculating combinations with repetition is given by $$C(n+r-1, r)$$, where n is the number of distinct items and r is the number of selections to be made.
2. In combinations with repetition, the same item can appear multiple times in a selection, which is different from regular combinations where each item can only appear once.
3. This concept is commonly used in real-life situations like making ice cream sundaes, where you can choose multiple scoops of the same flavor.
4. The problem of distributing r identical objects into n distinct boxes can be solved using combinations with repetition.
5. Understanding combinations with repetition is key for solving more complex problems involving multisets and various types of distributions.

Review Questions

• How would you apply the concept of combinations with repetition to solve a real-world problem such as creating a fruit salad?
  • When creating a fruit salad, if you want to choose 5 fruits from a selection of 3 types (apples, bananas, oranges), you can select each type multiple times. Using combinations with repetition allows you to find out how many different ways you can pick these fruits, regardless of order. This means you could have options like 2 apples, 2 bananas, and 1 orange as one possible combination.
• Compare and contrast combinations with repetition and permutations. What are the key differences between them?
  • Combinations with repetition allow for selections where items can be chosen more than once without regard to order, while permutations involve arranging items where order matters and each item is used exactly once. For example, choosing 3 fruits from 5 types allows repeats in combinations but not in permutations. The calculations for these two concepts also differ; combinations use a specific formula that includes repetitions, while permutations focus solely on arrangements.
• Evaluate a complex problem involving combinations with repetition: If you have 4 different types of candy and you want to create a box containing 10 pieces total, how would you calculate the number of ways to do this?
  • To solve this problem, we would use the stars and bars theorem since we want to distribute 10 indistinguishable pieces (stars) among 4 distinguishable types of candy (boxes). The calculation involves finding the number of solutions to the equation $$x_1 + x_2 + x_3 + x_4 = 10$$ where each $$x_i$$ represents how many pieces of each type are included. The formula would be $$C(10 + 4 - 1, 4 - 1) = C(13, 3)$$ which gives us the total number of ways to create that box of candy.

Related terms

Combinations: Selections made from a set where the order of items does not matter and each item can only be chosen once.

Permutations: Arrangements of items from a set where the order does matter, and each item is used exactly once.

Stars and Bars Theorem: A combinatorial method used to solve problems of distributing indistinguishable objects (like repetitions) into distinct boxes (like groups or selections).