Permutation
Who is 1st, 2nd, 3rd matters.
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GMAT Quant
Permutations and Combinations
Choose between arrangement and selection counting by deciding whether order changes the outcome.
Decision rule
Start with one question: does order matter?
If swapping selected items creates a different outcome, use a permutation. If the same group remains the same outcome, use a combination.
Lineups and groups are different counting problems.
Combination
Only who is selected matters.
Permutation: choose and arrange.
- Formula
- nPr = n!/(n-r)!
- Example
- 6P3
- Calculation
- 6*5*4
- Result
- 120
Combination: choose only.
- Formula
- nCr = n!/[r!(n-r)!]
- Example
- 6C3
- Calculation
- 120/6
- Result
- 20
Why it works
Why combinations divide by r!.
A chosen group of r people can be internally ordered r! ways. Those orders are duplicates when order does not matter.
Repeated items
Repeated identical items require another duplicate correction.
For n items with repeated groups of sizes a, b, c, ... use n!/(a!b!c!...).
Recognition
Common wording can reveal the model.
- 01Arrange, rank, seat, schedule, password -> order usually matters.
- 02Choose, select, committee, team, group -> order usually does not matter.
- 03If repetition is allowed, do not automatically use the no-repeat formulas.
Mini application
Try this: From 7 people, how many 3-person teams can be formed?
Decide the model before calculating.
Solution
The answer is 7C3 = 35.
A team has no positions, so different selection orders represent the same team.
Remember this
Permutation counts different orders; combination removes them.
When uncertain, swap two selected items mentally. If the outcome changes, order matters.