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Five runners need to be arranged in 5 lanes for a race so that there is a runner in each lane. How many unique ways are there to arrange the runners in the 5 lanes?

Five runners need to be arranged in 55 lanes for a race so that there is a runner in each lane.\newlineHow many unique ways are there to arrange the runners in the 55 lanes?

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Full solution

Q. Five runners need to be arranged in 55 lanes for a race so that there is a runner in each lane.\newlineHow many unique ways are there to arrange the runners in the 55 lanes?
  1. Problem Understanding: Understand the problem. We need to find the number of unique arrangements for 55 runners in 55 lanes, with one runner per lane.
  2. Permutation Recognition: Recognize that this is a permutation problem because the order in which we place the runners in the lanes matters.
  3. Permutation Formula: Use the formula for permutations, which is n!n! (nn factorial) when arranging nn distinct objects into nn places. Here, nn is 55 because there are 55 runners and 55 lanes.
  4. Factorial Calculation: Calculate the factorial of 55, which is 5!=5×4×3×2×15! = 5 \times 4 \times 3 \times 2 \times 1.
  5. Final Conclusion: Perform the calculation: 5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120.
  6. Final Conclusion: Perform the calculation: 5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120. Conclude that there are 120120 unique ways to arrange the five runners in the five lanes.

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