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← Unit 9 · Permutations vs combinations

Lesson

Read the rule, then work each example: try the setup yourself, then tap to check one step at a time.

The rules

Counting: which tool? Ask these 4 questions in order

Order? → pools? → cases? → repeats? Answer each before you touch the calculator.

  1. 1. Does ORDER matter? Swap two picks — new outcome? Yes → Permutation. No → Combination.
  2. 2. Separate POOLS with quotas (2 men AND 2 women)? → one count per pool, then MULTIPLY.
  3. 3. Several acceptable CASES ('or', 'at least', 'at most')? → count each case, then ADD (or total − the cases you don't want).
  4. 4. Can items REPEAT? Yes (digits, letters of an alphabet) → nʳ. No (people, cards, letters of a WORD) → n × (n − 1) × …

Swap test: permutation or combination?

Swap two of the chosen items — if you get a NEW outcome, order matters (permutation); if it's the same group, it's a combination.

P(n, r) and C(n, r) (nPr and nCr, or (n choose r))

  1. Permutation words: titles, 1st/2nd/3rd, codes, passwords, a name, a house number, pictures on a wall, 'assign to different tasks'.
  2. Combination words: committee, team, a selection of books, a collection of DVDs, hand of cards, 'select / choose' with no roles.
  3. Same 4 students: same task → C(12,4) = 495; different tasks → P(12,4) = 11,880.

P(n, r), C(n, r) and why ÷ r!

P(n, r) = n! ÷ (n − r)! = r factors counting down from n; C(n, r) = P(n, r) ÷ r!.

  1. AB and BA are 2 permutations but 1 combination; every group of r shows up r! times → divide by r!.
  2. C(n,1) = n; C(n,r) = C(n,n−r): C(11,7) = C(11,4) = 330.
  3. C(10,3) = (10·9·8) ÷ (3·2·1) = 120.

Multiply within a case, add across cases

AND = steps that build ONE outcome → ×. OR = separate outcomes with no overlap → +.

  1. List the acceptable cases (2W2M, 3W1M, 4W …).
  2. Inside each case multiply one count per pool.
  3. Add the cases.
  4. Check with total − unwanted cases when 'at least' is involved.
  5. Words CVV + VCV + VVV: 7·5·5 + 5·7·5 + 5·5·5 = 475.

Arranging with repeated items

Arrange n items with r₁, r₂, … identical copies: n! ÷ (r₁! r₂! …).

  1. AAAB → 4!/3! = 4.
  2. Only the arrangement of ALL the items uses this rule; choosing some of them is a combination problem.
Traps and calculator keys are on the unit page →

Counting: which tool?

Walk a problem through the 4 questions from your cheat sheet.

Which counting tool?

Step 1 of 4

    1 · Does ORDER matter?

    Swap test: swap two of the picks. Is it a NEW outcome?

    Yes: titles (president / VP), 1st / 2nd / 3rd place, codes, passwords, a name, pictures on a wall, "assign to different tests". No: committees, teams, a selection of books, a collection of DVDs, hands of cards, "select / choose".

    Worked examples (18)

    Example 1 · 2.5 #1a0/4

    List all permutations of 2 letters taken out from the 3 above.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 2 · 2.5 #1b0/4

    Use the formula to check your answer.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 3 · 2.5 #1c0/4

    List all permutations of 3 letters taken out from the 3 above.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 4 · 2.5 #1d0/4

    Use the formula to check your answer.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 5 · 2.5 #2a0/4

    List all permutations of these 4 letters

    ___

    Given 4 letters A, A, A, B.

    Try the setup yourself first, then tap.

    Example 6 · 2.5 #2b0/4

    Use the formula to check your answer.

    ___

    Given 4 letters A, A, A, B.

    Try the setup yourself first, then tap.

    Example 7 · 2.5 #3a0/4

    List all permutations of these 4 letters

    ___

    Given 4 letters A, A, B, B

    Try the setup yourself first, then tap.

    Example 8 · 2.5 #3b0/4

    Use the formula to check your answer.

    ___

    Given 4 letters A, A, B, B

    Try the setup yourself first, then tap.

    Example 9 · 2.5 #4a0/4

    List all combinations of 2 letters taken out from the 3 above.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 10 · 2.5 #4b0/4

    Use the formula to check your answer.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 11 · 2.5 #4c0/4

    List all combinations of 3 letters taken out from the 3 above.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 12 · 2.5 #4d0/4

    Use the formula to check your answer.

    ___

    Given 3 letters A, B, C.

    Try the setup yourself first, then tap.

    Example 13 · 2.5 #5a0/5

    A name

    ___

    Define the object in each arrangement, and then determine if it's a Combination or a Permutation.

    Try the setup yourself first, then tap.

    Example 14 · 2.5 #5b0/5

    A hiring committee

    ___

    Define the object in each arrangement, and then determine if it's a Combination or a Permutation.

    Try the setup yourself first, then tap.

    Example 15 · 2.5 #5c0/5

    A house number

    ___

    Define the object in each arrangement, and then determine if it's a Combination or a Permutation.

    Try the setup yourself first, then tap.

    Example 16 · 2.5 #5d0/5

    A collection of DVDs

    ___

    Define the object in each arrangement, and then determine if it's a Combination or a Permutation.

    Try the setup yourself first, then tap.

    Example 17 · 2.5 #5e0/5

    An arrangement of pictures on the wall

    ___

    Define the object in each arrangement, and then determine if it's a Combination or a Permutation.

    Try the setup yourself first, then tap.

    Example 18 · 2.5 #5f0/5

    A selection of books

    ___

    Define the object in each arrangement, and then determine if it's a Combination or a Permutation.

    Try the setup yourself first, then tap.

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