An elevator starts with 5 passengers and stops independently at each of 8 floors. Assuming each passenger independently chooses a floor uniformly among the 8, what is the probability that all 5 alight at different floors?

Introduction / Context:
We model each passenger’s floor choice as an independent uniform pick among 8 floors. We seek the probability that the 5 chosen floors are all distinct.

Given Data / Assumptions:

• Passengers choose floors independently and uniformly from {1,…,8}.
• Choices are with replacement (conceptually), since multiple people could choose the same floor.

Concept / Approach:
Favourable outcomes: permutations of choosing 5 distinct floors from 8 → P(8,5) = 8*7*6*5*4. Total outcomes: 8^5 ordered choices. Probability = P(8,5)/8^5.

Step-by-Step Solution:
P = (8*7*6*5*4) / 8^5 = (8/8)*(7/8)*(6/8)*(5/8)*(4/8).= 1 * (7/8) * (3/4) * (5/8) * (1/2) = (105)/(512).

Verification / Alternative check:
Compute numerators and denominators explicitly: 8*7*6*5*4 = 6720; 8^5 = 32768; 6720/32768 reduces to 105/512.

Why Other Options Are Wrong:
Other fractions correspond to nearby but incorrect numerators in the product (e.g., replacing 6/8 by 1/2 too early or arithmetic slips).

Common Pitfalls:
Using combinations instead of permutations; order matters because passengers are distinguishable.

Final Answer:
105/512