Step by Step
1
Permutations ā order matters
Use permutations when the order of selection matters ā like passwords, race results (1st, 2nd, 3rd place), or seating arrangements. Formula: nPr = n! / (nār)!.
2
Combinations ā order doesn't matter
Use combinations when only the selection itself matters, not the order ā like committees, card hands, or pizza toppings. Formula: nCr = n! / [r!(nār)!].
3
The key diagnostic question
Ask yourself: if I rearranged this same selection into a different order, would it count as a different outcome? If yes, use permutations. If no (it's the same outcome regardless of order), use combinations.
4
How the two formulas relate
nCr = nPr / r! ā combinations are permutations divided by the number of ways to rearrange each group internally (r!), since combinations don't care about that internal ordering.
Applied Walkthrough
1
A club with 10 members needs to choose a President, Vice President, and Treasurer (three distinct roles). Since the roles are distinct ā being President is different from being Treasurer ā order matters. Use permutations: 10P3 = 10!/(10-3)! = 10Ć9Ć8 = 720.
2
Now the same club needs to choose a 3-person committee, where all three members have equal, undifferentiated roles. Since there's no distinction between the members once chosen (it's just "the committee"), order doesn't matter. Use combinations: 10C3 = 10!/[3!(10-3)!] = 120.
3
Notice: 720/120 = 6, and 3! = 6 ā confirming the relationship nCr = nPr/r!, since there are 3! = 6 ways to rearrange any specific group of 3 people internally.
4
This distinction ā distinct roles (permutation) versus an undifferentiated group (combination) ā is the exact diagnostic question to ask on any counting problem.
Exam Application
Exams test whether you can correctly determine whether a scenario calls for a permutation or combination by asking whether order/distinct roles matter, and whether you can apply the correct formula (nPr or nCr).
ā Common Trap
The most common trap is failing to ask the key diagnostic question (does rearranging change the outcome?) and instead guessing based on surface features of the problem ā always explicitly check whether the roles or order matter before choosing a formula.
ā Quick Self-Check
1. When should you use a permutation instead of a combination?
When the order of selection matters (distinct roles or positions).
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2. What is the formula for permutations?
nPr = n! / (nār)!.
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3. What is the formula for combinations?
nCr = n! / [r!(nār)!].
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4. For a club choosing a President, VP, and Treasurer from 10 members, would you use a permutation or combination, and why?
Permutation, since the three roles are distinct, so order/role assignment matters.
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5. How does nCr relate to nPr?
nCr = nPr / r! ā combinations divide out the r! ways to rearrange each selected group internally.
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