Combinations - GMAT Math Study Guide
Definitions
- Combinatorics - The branch of mathematics that deals with collections of objects that satisfy specified criteria (e.g., counting arrangements, permutations, and combinations).
For example, combinatorics would answer the question "how many different ways can 5 men be seated at a circular table?"
- Combinations - The branch of combinatorics where changing the order of the objects does not create a new scenario.
For example, the question "how many different ways can one compose a team of 5 co-workers from a pool of 10 employees?" is a combinations question since changing the order in which the 5 employees are picked does not create a new arrangement.
- Permutations - The branch of combinatorics where changing the order of the objects creates a new scenario.
For example, the question "how many different ways can the team rank in the tournament?" is a permutations question because a team moving from first to second place creates a new arrangement (see permutations for much more on this topic).
Combinations - Defined by Example
Although the above definitions likely provided some clarity about combinations, the concept of combinatorics and combinations vs. permutations can be confusing. Consequently, we will consider an example in detail to help clarify.
An Example Problem
An accounting firm recently won a large and high-value client. The partner at the accounting firm in charge of conducting the audit needed to pick a team of 3 individuals to work on the account from a pool of 5 employees. How many different teams can be formed?
Combination
The above scenario would normally be read as a combinations problem (and not a permutations problem) since changing the order in which the accountants were selected would not create a new possible team.
Pool of Employees to Draw From: A, B, C, D, E
Possible Scenario: 10
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
As a point of clarification, choosing BAC would not create a new scenario under combinations (although it would under permutations).
Determining the number of scenarios does not have to be done this long way. A shorter formula that uses factorials exists and is discussed below.
Permutation
The above scenario would become a permutations scenario if changing the order in which the 3 individuals who composed the team were chosen created a new arrangement (e.g., ABC and ACB counted as two separate possible arrangements). Since this is not the natural reading of the scenario, the text describing the example scenario would need to be changed to reflect this (i.e., to make it a permutations problem).
Combinations
Combinations problems such as the one involving the selection of 3 individuals from a group of 5 do not need to be solved by drawing out all the possible scenarios. The generic formula for selecting k objects from a pool of n objects when the order in which the n objects is chosen does not matter is given by the following formula:
Combinations Formula
Examples
Both permutations and combinations are best learned through working examples.
How many different groups of 10 students can a teacher select from her classroom of 15 students?
Step 1: Determine whether the question pertains to permutations or combinations.
Since changing the order of the selected students would not create a new group, this is a combinations problem.
Step 2: Determine n and k
n = 15 since the teacher is choosing from 15 students
k = 10 since the teacher is selecting 10 students
Step 3: Apply the formula
Another Example
In order to form a co-ed water polo team to compete in a local business league, a manager must select 3 men and 4 women from among 4 men and 6 women. How many different combinations of teams can the manager form?
Step 1: Determine whether the question pertains to permutations or combinations.
Since changing the order in which the team members are selected does not produce a new arrangement, this is a combinations problem.
Step 2: Determine n and k
This problem is more complicated than the previous one due to the inclusion of two sub-groups among those selected (i.e., men and women).
The total number of combinations = (ways to select 3 men)*(ways to select 4 women)
For Men: n = 4, k = 3
For Women: n = 6, k = 4
Written in formula form:
4C
3 *
6C
4
Step 3: Apply the formula
