Combinations Fundamentals

A combination is a selection of objects where order does not matter. Two selections that use the same objects, regardless of the order they were picked in, count as the same combination. This lesson builds the combination formula directly from the permutation formula and gives a clear test for choosing between the two.

What you will learn

• Define a combination as an unordered selection
• Apply the combination formula C(n,r)
• Distinguish when to use permutations versus combinations

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

What is a combination?

A combination is an unordered selection of some of the objects from a larger set. If a club with members Ana, Ben, Cy, and Dee needs to pick a 2-person committee where both members have the identical role, then picking Ana and Ben is the exact same committee as picking Ben and Ana — the order the two names were chosen in makes no difference to the outcome. This is the opposite situation from a permutation, where swapping two objects always creates a new, distinct arrangement.

Listing every 2-person committee from these 4 members directly shows this: AB, AC, AD, BC, BD, and CD are the only distinct pairs, giving 6 committees in total. Notice that BA was never listed separately from AB, because the two labelings describe the same committee.

From permutations to combinations

The permutation count $P(n,r)$ counts ordered selections of $r$ objects from $n$, so it treats every reordering of the same $r$ objects as a different outcome. Each unordered group of $r$ objects can itself be arranged in $r!$ different orders, so every combination gets counted exactly $r!$ times inside $P(n,r)$.

Dividing removes that overcounting: the number of combinations is $\binom{n}{r}=\dfrac{P(n,r)}{r!}=\dfrac{n!}{r!\,(n-r)!}$. The symbol $\binom{n}{r}$ is read "n choose r" and is the standard notation for a combination count.

Choosing between permutations and combinations

Certain words signal an unordered selection: "choose," "select," or "form a group, committee, or team" where the members share one role. Other words signal an ordered selection: "arrange," "order," "assign to positions," or "line up," or any phrasing where the roles being filled are different from each other.

The same group of people can give two different counts depending on the roles involved. Choosing 2 co-captains from a team of 6, where both co-captains have the identical title, is a combination: $\binom{6}{2}=15$. Choosing a president and a vice president from the same 6 people, where the two titles are different, is a permutation: $P(6,2)=30$, exactly double the combination count, because each pair of people can be split into the two distinct roles in 2 ways.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Using the permutation count $P(n,r)$ when the problem describes an unordered group, which overcounts every selection by $r!$.
• Dividing by the wrong factorial, such as dividing by $(n-r)!$ a second time instead of by $r!$.
• Forgetting to reduce the fraction before multiplying, leading to arithmetic errors with unnecessarily large numbers.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• A combination counts unordered selections; swapping two chosen objects gives the same outcome.
• The combination formula divides the permutation count by $r!$: $\binom{n}{r}=\dfrac{n!}{r!\,(n-r)!}$.
• Decide between permutations and combinations by checking whether swapping two chosen objects changes the roles involved.