Combinations with Repetition – Selecting with Repeats Allowed
Ordinary combinations assume you can only select each item once. But sometimes you can pick the same option more than once and order still doesn't matter – like choosing 3 scoops of ice cream from 5 flavours, where “chocolate, chocolate, vanilla” is a perfectly valid choice. The formula for this is C(n + r − 1, r), where n is the number of options and r is how many you're choosing.
This formula looks unusual at first, but it comes from a clever trick called “stars and bars”: imagine r stars (your r selections) separated by dividers marking which flavour each group belongs to, and count the ways to arrange those stars and dividers together.
The Repetition Combination Formula
Combinations of r items chosen from n options, with repetition allowed and order not mattering: C(n + r − 1, r).
C(4 + 3 − 1, 3) = C(6, 3) = 20.
Real-Life Application
- Grocery shopping: choosing multiple units of the same product from a range of items.
- Investment portfolios: choosing how to distribute funds among a set of investment types.
- Recipe scaling: choosing how many of each of several ingredients to include.
Key Takeaways
- Combinations with repetition allow the same item to be chosen more than once.
- The formula is C(n + r − 1, r).
- The formula comes from the “stars and bars” counting trick.
Practice: Combinations with Repetition
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