Combinations – Unordered Selections
A combination counts the number of ways to choose r items from a set of n, where order does not matter – the opposite condition from a permutation. The formula is nCr = n! ÷ (r! × (n − r)!), which is exactly the permutation formula divided by r!, since every unordered group of r items corresponds to r! different ordered arrangements that all count as the same combination.
The clue that a problem needs combinations rather than permutations: does swapping the order of your selection make a different outcome? Choosing a committee of Alex, Sam, and Jo is the same committee no matter what order you name them in – that's a combination.
The Combination Formula
nCr = n! ÷ (r! × (n − r)!). This is nPr divided by r!, correcting for the r! orderings that all count as one combination.
6C2 = 6! ÷ (2! × 4!) = (6 × 5) ÷ (2 × 1) = 30 ÷ 2 = 15.
Real-Life Application
- Lottery draws: counting how many possible ticket combinations exist.
- Committee selection: counting how many different committees can be formed from a group.
- Card games: counting how many possible hands can be dealt.
Key Takeaways
- A combination counts unordered selections, where order doesn't matter.
- nCr = n! ÷ (r! × (n − r)!).
- Use combinations whenever swapping the order gives the same outcome.
Practice: Combinations
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