Every combinatorics question on the GMAT involves one or more of three fundamental combinatorics tasks: ordering, permutations, and combinations. This article will introduce the three tasks, and each of the three following articles will explain a task intuitively to arrive at the standard formula used when completing that task.
The Ordering Task
The ordering task is “lining up” elements spatially or temporally (in time).
- Spatial: How many ways can 7 cars be lined up in 7 parking spaces?
- Temporal: In how many different orders can 7 contestants perform in a talent show?
There is no mathematical difference between these two scenarios. Both can be represented with the same standard formula.
The Permutations Task: Selecting and Ordering
The permutations task is “lining up” a subgroup of elements spatially or temporally. We may refer to permutations as “selecting and ordering.” We will order elements in time or space, but only the subgroup of elements that has been selected out of the larger group. As with the ordering task, spatial and temporal instances of the permutations task are mathematically identical.
- Spatial: How many ways can 4 cars from a collection of 7 cars be lined up in 7 parking spaces?
- Temporal: The judges of a talent show will create a performance schedule for 4 finalists selected from among 7 semifinalists. How many different schedules can be created?
The Combinations Task: Selecting Without Ordering
The combinations task is selecting a subgroup of elements out of a larger group, without regard for their order. We may refer to combinations as “selecting without ordering,” or simply “selecting.” In combinations, the spatial/temporal distinction disappears altogether. Since nothing is being “lined up,” there is no time or space where this “lining up” occurs. Selecting a subgroup out of a larger group is not a spatial or temporal operation.
- Spatial: How many ways can 4 cars be selected from a collection of 7 cars?
- Temporal: How many ways can 4 finalists be chosen from among 7 semifinalists in a talent show?
Conclusion
With all three tasks in view, it makes sense to think of permutations as “selecting, then ordering.” Permutations entails performing the combinations task and then performing the ordering task on the selected subgroup. In the coming articles, we will see how the standard formulas for each task relate and support this understanding.
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Contributor: Elijah Mize (Apex Instructor)