Math101 · Independent Ontario learning library
Math101learn.math101.caCombinations
Combinations count unordered selections, where only membership matters and rearranging the same chosen objects does not create a new outcome.
Use a combination when the question asks who or what is selected, not the order in which selections appear.
Formula
The number of ways to choose $r$ objects from $n$ distinct objects without order is
The conditions are $0\le r\le n$. Choosing no objects or all objects each has one outcome.
Why divide by r factorial
The permutation count ${}_nP_r$ counts every selected group in all $r!$ possible internal orders. A combination treats those orders as the same group, so
This relationship explains the denominator rather than making it a mysterious formula.
Worked example
The same four students listed in another order remain the same committee.
Combination or permutation?
Ask whether swapping positions creates a different outcome. Choosing three pizza toppings is a combination; assigning gold, silver, and bronze among three people is a permutation.
Named roles can turn a selection problem into a two-stage process: first choose members, then assign roles.
Required members
If two particular students must be on a five-person committee, include them and choose the remaining three from everyone else.
The required members are not chosen again. Translate restrictions before entering numbers into a formula.
At least and at most conditions
Split possibilities into disjoint cases and add. “At least two” might mean exactly two, exactly three, and so on.
Alternatively, use a complement when few excluded cases exist:
Choosing from categories
To select $2$ students from one class and $3$ from another, multiply independent category choices:
If several category compositions are allowed, count each disjoint composition and add the results.
Symmetry identity
Choosing $r$ objects to include is equivalent to choosing $n-r$ objects to leave out:
This can simplify calculations and provides an intuitive check.
Binomial connection
Combination values are coefficients in
They also count which $r$ trials are successes in a binomial probability model.
Probability applications
When drawing without replacement and order is irrelevant, combinations often count equally likely subsets. For example, a five-card hand is a subset, not an ordered deal, if only the final hand matters.
Use the same outcome definition in favourable and total counts.
Common mistakes
Using combinations for assigned roles. Roles create order.
Using permutations and forgetting to divide by internal orders. The same group gets overcounted.
Multiplying cases that are alternatives. Disjoint “or” cases are added.
Using complement counts without defining the universe. Start from the correct total set.
Mixing ordered favourable outcomes with unordered total outcomes. Count consistently.
Quick self-check
- Does rearranging the same selected members change the outcome?
- What are $n$ and $r$ after restrictions are applied?
- Are required or forbidden members handled before the formula?
- Should category counts multiply or case counts add?
- Would a complement be shorter?
- Is the probability sample space counted with the same order convention?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
How many 4-person committees can be chosen from 10 students?
- 10C4 = 10!/(4!6!)
- 10C4 = 210.
End of lesson
Nice work making it this far.
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