How Many Ways Can Four People Finish a Race? Permutations Explained

Understanding Permutations vs. Combinations

• 🔑 In permutations, the order matters, meaning different arrangements of the same items are counted as distinct.
• 💡 For combinations, the order does not matter; only the selection of items is considered, regardless of arrangement.
• 🧩 Example: The letters ABC can be arranged as ABC and ACB. These are two permutations but one combination because the same three letters are used.

Solving the Race Problem with Permutations

• 🎯 The question "In how many different ways can four people finish a race?" indicates a permutation problem because the order of finishing is crucial.
• ⚡ Method 1: Fundamental Counting Principle
  • There are 4 choices for first place.
  • Once first place is decided, there are 3 remaining choices for second place.
  • Then, there are 2 choices for third place.
  • Finally, there is 1 choice left for fourth place.
  • The total number of ways is calculated by multiplying these possibilities: 4 * 3 * 2 * 1 = 24.
• 🧮 Method 2: Permutation Formula
  • The formula is nPr = n! / (n-r)!, where n is the total number of items and r is the number of items to choose.
  • In this case, n=4 (four people) and r=4 (all four finish the race).
  • So, it's 4P4 = 4! / (4-4)! = 4! / 0!.
  • Since 4! = 24 and 0! = 1, the result is 24 / 1 = 24.

Conclusion

• 🏆 Both methods confirm that there are 24 different ways four people can finish a race.