Combinations of r Objects from n
Simple Explanation
A combination is an UNORDERED selection — choosing r objects from n, where the order the objects were chosen in does not matter. The count is C(n,r) = n! / [r!(n−r)!].
Why Do We Need It?
Many real selection problems genuinely do not care about order (like choosing a committee, where being "chosen first" carries no special meaning) — combinations are the correct tool for these cases.
Formula
Combinations of r Objects from n
C(n,r) = n! / [r!(n−r)!]
The number of ways to choose r objects from a total of n distinct objects, where the ORDER of selection does not matter.
- n
- — the total number of distinct objects available
- r
- — the number of objects being chosen, r ≤ n
When to use it: Whenever a group or subset is being chosen and the order within the group does not matter (like choosing a committee).
Worked Example
Count an unordered selection
In how many ways can a committee of 3 people be chosen from a group of 8 people?
Why Does This Work?
Starting from P(n,r) (which DOES care about order), each unordered group of r objects has been counted r! times — once for every possible ordering of that same group. Dividing by r! removes this extra overcounting, leaving only the number of genuinely distinct, unordered groups.
Real-Life Example
Choosing lottery numbers or a committee
A lottery draw selects a set of numbers, or an organization selects a committee — in both cases, the ORDER of selection does not matter, only which items ended up chosen.
Combinations give the correct count of possible outcomes in both these common real-world selection scenarios.
Practice
In how many ways can 2 people be chosen from a group of 6 to represent the group at a conference (order does not matter)?
MediumCommon mistake
Applying the permutations formula instead, when order truly does not matter (like choosing a committee, not ranking its members) — this overcounts by a factor of r!.
Quick Review
- C(n,r) = n! / [r!(n−r)!].
- Use whenever ORDER does NOT matter among the r chosen objects.
- Related to permutations by dividing out the r! orderings of each group.