**GRE Advanced Counting: Permutations vs. Combinations – When Order Matters**

GRE Advanced Counting: Permutations vs. Combinations – When Order Matters

A Premium Study Guide for Serious GRE Candidates

What This Is

The GRE tests counting problems—questions that ask, "In how many ways can X happen?"—to assess your ability to distinguish between permutations (order matters) and combinations (order doesn’t matter). These questions appear in Quantitative Reasoning (usually 1–2 per test) and can boost your score by 5–10 points if mastered. They’re deceptively simple but riddled with traps (e.g., overcounting, misapplying formulas). Example:

A password consists of 4 distinct letters from the English alphabet. How many different passwords are possible? (A) 26 × 25 × 24 × 23 (B) 26! / (22! × 4!) (C) 26! / 22! (D) 26 × 25 × 24 × 23 / 4!

Why this matters: The difference between (A) and (B) is order. (A) counts permutations (ABCD ≠ BACD), while (B) counts combinations (ABCD = BACD). The GRE loves testing this distinction.

Key Concepts & Techniques

1. The Fundamental Counting Principle (FCP)
2. What it is: If Event A can happen in m ways and Event B in n ways, the total ways both can happen is m × n.
3. When to use: For sequential choices (e.g., picking a shirt then pants, or assigning roles to people).

4. Example: A 3-digit code with no repeats: 10 × 9 × 8 = 720 ways.

5. Permutations (Order Matters)

6. What it is: Arrangements where sequence matters (e.g., passwords, rankings, seating).
7. Formula: P(n, k) = n! / (n – k)! (or n × (n–1) × ... × (n–k+1)).
8. When to use: When the problem says "arrange," "order," "rank," "assign distinct positions," or implies uniqueness (e.g., "first place, second place").

9. Example: 5 runners in a race: P(5, 3) = 5 × 4 × 3 = 60 ways to award gold, silver, bronze.

10. Combinations (Order Doesn’t Matter)

11. What it is: Selections where sequence is irrelevant (e.g., teams, committees, groups).
12. Formula: C(n, k) = n! / (k! × (n – k)!).
13. When to use: When the problem says "choose," "select," "group," "combination," or implies unordered sets (e.g., "a committee of 3").

14. Example: 10 people, choose 3 for a team: C(10, 3) = 120.

15. Slot Method (For Permutations)

16. What it is: Draw k slots (_ _ ____), fill each with the number of available choices, then multiply.
17. When to use: For permutation problems with restrictions (e.g., "no repeats," "must include X").

18. Example: 4-letter password, no repeats, must include "A": Fix "A" in one slot (4 choices for position), then fill the rest: 4 × 25 × 24 × 23.

19. Anagram Method (For Combinations)

20. What it is: Treat the problem as an anagram of n items where k are identical (e.g., YYYNNN for "yes" and "no" choices).
21. When to use: For combination problems with identical items (e.g., "select 3 identical books from 5").

22. Example: 7 books, 3 identical math books, 4 identical history books. Choose 4: Anagram of MMMHHHH → C(7, 3) = 35 (since M’s and H’s are identical).

23. Complementary Counting

24. What it is: Calculate the total ways, then subtract the unwanted cases.
25. When to use: When the problem asks for "at least one" or "not all" (e.g., "at least one woman on a committee").

26. Example: 5 men, 4 women. Committee of 3 with at least 1 woman: Total = C(9, 3) = 84. Unwanted = C(5, 3) = 10. Answer = 84 – 10 = 74.

27. Symmetry in Combinations

28. What it is: C(n, k) = C(n, n–k). Useful for simplifying calculations.
29. When to use: When k is large (e.g., C(10, 7) = C(10, 3)).

30. Example: C(12, 9) = C(12, 3) = 220.

31. Circular Permutations

32. What it is: Arrangements in a circle (e.g., people around a table). Formula: (n–1)!.
33. When to use: For circular arrangements (e.g., "seating around a round table").
34. Example: 5 people around a table: 4! = 24 ways.

Step-by-Step Strategy

Follow these steps for every counting problem:

1. Read the question carefully. Underline:
2. Keywords ("arrange," "choose," "order," "select").
3. Restrictions ("no repeats," "must include X," "at least one").

4. Whether the answer is a permutation (order matters) or combination (order doesn’t).

5. Decide: Permutation or Combination?

6. Permutation: If the problem involves ranking, assigning distinct positions, or sequences (e.g., passwords, race results, seating in a row).

7. Combination: If the problem involves groups, teams, or unordered selections (e.g., committees, handshakes, identical items).

8. Apply the correct formula or method.

9. Permutation: Use P(n, k) or the slot method.
10. Combination: Use C(n, k) or the anagram method.

11. Restrictions: Use complementary counting or fix-and-fill (e.g., "must include X").

12. Check for overcounting or undercounting.

13. Overcounting: Are you counting ABC and BAC as different when they shouldn’t be? (Use combinations.)

14. Undercounting: Are you missing cases where order matters? (Use permutations.)

15. Calculate and match to the answer choices.

16. Simplify fractions early (e.g., 10! / 7! = 10 × 9 × 8).

17. Look for symmetry (e.g., C(10, 7) = C(10, 3)).

18. Verify with a smaller example.

19. Test your logic with smaller numbers to ensure it’s correct.

Fully Worked Example

GRE-Style Question:
A club has 8 members. In how many ways can they elect a president, vice president, and treasurer if no member can hold more than one position?

Step 1: Underline keywords.
- "elect a president, vice president, and treasurer" → distinct positions (order matters).
- "no member can hold more than one position" → no repeats.

Step 2: Permutation or Combination?
- Permutation (order matters: president ≠ vice president).

Step 3: Apply the formula.
- P(8, 3) = 8 × 7 × 6 = 336.

Step 4: Check for overcounting.
- No overcounting (each role is distinct).

Step 5: Match to answer choices.
- (A) 8 × 7 × 6 = 336 - (B) 8! / 5! = 336 - (C) C(8, 3) = 56 - Correct answer: (A) or (B).

Step 6: Verify with a smaller example.
- 3 members, elect 2 roles: P(3, 2) = 6. List them: AB, BA, AC, CA, BC, CB → 6 ways. Correct.

Common Mistakes

1. Mistake: Using C(n, k) when order matters.
2. Why it happens: Students see "choose" and default to combinations.

3. Correct approach: Ask: "Does swapping two items create a new valid arrangement?" If yes → permutation.

4. Mistake: Forgetting to divide by k! in combinations.

5. Why it happens: Misapplying the permutation formula to combination problems.

6. Correct approach: C(n, k) = P(n, k) / k!. Always divide by k! when order doesn’t matter.

7. Mistake: Overcounting identical items.

8. Why it happens: Treating identical items as distinct (e.g., counting "AAB" and "ABA" as different when they’re the same).

9. Correct approach: Use the anagram method for identical items.

10. Mistake: Ignoring restrictions (e.g., "must include X").

11. Why it happens: Rushing and missing key details.

12. Correct approach: Fix the restricted item first, then fill the rest (e.g., "must include A" → fix A in one slot, then arrange the others).

13. Mistake: Misapplying complementary counting.

14. Why it happens: Forgetting to subtract the unwanted cases from the total.
15. Correct approach: Total = C(9, 3). Unwanted = C(5, 3). Answer = Total – Unwanted.

GRE Traps & Timing

1. Trap: "At least one" vs. "Exactly one"
2. "At least one woman" = Total – All men.
3. "Exactly one woman" = C(women, 1) × C(men, 2).

4. How to avoid: Circle the wording. "At least" = complementary counting.

5. Trap: Hidden order

6. "A team of 3 where one is captain" → order matters (captain is a distinct role).

7. How to avoid: Ask: "Is one position special?" If yes → permutation.

8. Trap: Repeated elements

9. "How many 4-letter words can be made from MISSISSIPPI?" → identical letters (e.g., S’s are identical).

10. How to avoid: Use the anagram method (count total letters, divide by repeats).

11. Timing:

12. Easy question: 60–90 seconds.
13. Hard question (with restrictions): 90–120 seconds.
14. Pro tip: If stuck, plug in smaller numbers to test your logic.

Quick Practice

1. A pizza shop offers 5 toppings. How many different 3-topping pizzas can be made?

2. Answer: C(5, 3) = 10 (order doesn’t matter).

3. How many ways can 5 books be arranged on a shelf if 2 are identical?

4. Answer: 5! / 2! = 60 (anagram method for identical items).

Last-Minute Cram Sheet

1. Permutation (order matters): P(n, k) = n! / (n–k)!
2. Combination (order doesn’t matter): C(n, k) = n! / (k! × (n–k)!)
3. Slot method: Draw blanks, fill with available choices, multiply.
4. Anagram method: Treat as an anagram (e.g., YYYNNN for "yes/no" choices).
5. Complementary counting: Total – Unwanted = Desired.
6. Symmetry: C(n, k) = C(n, n–k).
7. Circular permutations: (n–1)!.
8. Fix-and-fill: For restrictions (e.g., "must include X"), fix X first.
9. ⚠️ Trap: "At least one" ≠ "exactly one."
10. ⚠️ Trap: Identical items → divide by k! (e.g., n! / (k! × m!) for repeats).