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Counting Principles: Permutations and Combinations

Permutation (order matters): How many ways to arrange r items from n? Combination (order doesn't matter): How many ways to choose r items from n? Quick ru…

Before you read
Answer first — see what you already know.

How many ways can you arrange 4 letters from the word MATH?

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Answer:

C

Before you read
Answer first — see what you already know.

From 10 runners, how many ways can gold, silver, and bronze medals be awarded?

Check your answer
Answer:

B

Theory

Permutations vs Combinations

Permutation (order matters): How many ways to arrange rr items from nn?
P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n-r)!}

  1. P(n,r)=n!/(n−r)!P(n,r) = n!/(n-r)!
  2. P(5,3)=5!/(5−3)!P(5,3) = 5!/(5-3)!choose 3 from 5, order matters
  3. =5!/2!= 5!/2!
  4. =120/2=60= 120/2 = 60
Permutation: P(5,3) = 60 arrangements

Combination (order doesn't matter): How many ways to choose rr items from nn?
C(n,r)=(nr)=n!r!(n−r)!C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

  1. C(n,r)=n!/[r!(n−r)!]C(n,r) = n!/[r!(n-r)!]
  2. C(5,3)=5!/[3!⋅2!]C(5,3) = 5!/[3!·2!]choose 3 from 5, order doesn't matter
  3. =120/(6⋅2)= 120/(6·2)
  4. =120/12=10= 120/12 = 10
Combination: C(5,3) = 10 groups

Quick rule: "Order matters →\to Permutation. Order doesn't →\to Combination."

Example: Picking a president + VP from 10 people →\to P(10,2)=90P(10,2) = 90 (order matters).
Picking a committee of 2 from 10 →\to C(10,2)=45C(10,2) = 45 (order doesn't matter).

Example 1

How many ways can 3 books be arranged on a shelf from 8 books?

Order matters (arrangement) →\to Permutation

P(8,3)=8×7×6=336P(8,3) = 8 \times 7 \times 6 = 336

336
Example 2

How many ways can a committee of 3 be chosen from 8 people?

Order doesn't matter →\to Combination

C(8,3)=8!3!5!=3366=56C(8,3) = \frac{8!}{3!5!} = \frac{336}{6} = 56

56
Tip

ACT Pro Tip

The ACT tests permutations/combinations but the SAT doesn't. Key decision: ask 'Does the order of selection matter?' If picking roles (president, VP) →\to permutation. If picking a group (committee) →\to combination. When in doubt, think: 'Would rearranging the selected items create a different outcome?'

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