Combinations with Repetition

Key idea

Stars and bars.

In depth

Choosing r items from n types with repetition allowed (order ignored) has C(n + r − 1, r) options. The stars and bars method turns it into placing n − 1 dividers among r stars.

Worked example

Choosing 3 scoops from 5 flavours: C(7, 3) = 35.

Formula

Real-world uses

• Choosing snacks from a shop when several of the same item are allowed

Common mistake

Using C(n, r), which forbids repeats.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Discrete Math & Number Theory

• Strong Induction – Assume all earlier cases to prove the next.
• Well-Ordering Principle – Every nonempty set of positive integers has a least element.
• Partial Orders – Reflexive, antisymmetric, transitive relations.
• Cardinality of Sets – Comparing sizes of infinite sets.
• Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
• Cantor’s Diagonal Argument – Proof that real numbers are uncountable.