Mastering Permutations and Combinations
- The Golden Rule: Order Matters
The absolute easiest way to tell them apart is to ask yourself one question: Does the order of the items matter?
• Permutation = Order MATTERS. Think of it as Position or Placement.
(For example, A-B-C is different from C-B-A).
• Combination = Order DOES NOT matter. Think of it as a Committee or a Chosen group.
(For example, A-B-C is the exact same group as C-B-A).
- Core Differences and Formulas
Permutations (P)
• Concept: Arranging items in a specific order.
• Keyword Clues: Arrange, Line up, Schedule, Award (1st, 2nd, 3rd), Code, Password.
• Formula: P(n, r) = n! / (n - r)!
Combinations (C)
• Concept: Selecting items where order is completely ignored.
• Keyword Clues: Choose, Select, Group, Committee, Team, Handshake.
• Formula: C(n, r) = n! / [r! * (n - r)!]
Note: "n" is the total number of items available, "r" is the number of items you are choosing, and "!" means factorial (e.g., 4! = 4 * 3 * 2 * 1 = 24).
- Examples and Step-by-Step Solutions
Example 1: The Race (Permutation)
Question: There are 8 runners in a race. In how many ways can the gold, silver, and bronze medals be awarded?
• Why it's a Permutation: The order matters. Getting 1st place is completely different from getting 3rd place.
• Calculation: Here, n = 8 and r = 3.
P(8, 3) = 8! / (8 - 3)!
P(8, 3) = 8! / 5!
P(8, 3) = (8 * 7 * 6 * 5!) / 5!
P(8, 3) = 8 * 7 * 6
Answer = 336 ways
Example 2: The Pizza Toppings (Combination)
Question: A pizza parlor offers 10 different toppings. You want to choose 3 toppings for your pizza. How many different pizzas can you create?
• Why it's a Combination: The order does not matter. Putting pepperoni, mushrooms, and onions on a pizza is the exact same pizza as onions, mushrooms, and pepperoni.
• Calculation: Here, n = 10 and r = 3.
C(10, 3) = 10! / [3! * (10 - 3)!]
C(10, 3) = 10! / (3! * 7!)
C(10, 3) = (10 * 9 * 8 * 7!) / ((3 * 2 * 1) * 7!)
C(10, 3) = (10 * 9 * 8) / (3 * 2 * 1)
C(10, 3) = 720 / 6
Answer = 120 ways
- Quick Tips to Spot the Difference
• The "Change the Order" Test: Pick a random result from the problem (like two people chosen for a task, say Alex and Ben). Swap their positions. Does the meaning change?
If yes (e.g., Alex is President, Ben is VP vs. Ben is President, Alex is VP), it is a Permutation.
If no (e.g., Alex and Ben are on a cleaning team vs. Ben and Alex are on a cleaning team), it is a Combination.
• Look for Hierarchy: Any time a problem mentions specific roles, titles, specific seats, or distinct prizes, it automatically forces a strict order. Use Permutations.
• Combinations are Always Smaller: Because order doesn't matter in combinations, you divide by an extra r! to eliminate duplicate arrangements. Your answer for a combination problem will always be smaller than or equal to a permutation problem with the same numbers.
