An elevator starts with 5 passengers and stops at 8 different floors of the house. Find out the probability of all the 5 passengers alighting at different floors.

Options

A. a. `101/512`
B. b. `105/512`
C. c. `107/512`
D. d. `109/512`

Answer

To find the probability that all 5 passengers alight at different floors when an elevator stops at 8 different floors, we can follow these steps:

### Step 1: Determine the Total Outcomes
The total number of ways in which 5 passengers can alight on 8 different floors is calculated as follows:
- Each of the 5 passengers can choose any of the 8 floors.
- Therefore, the total number of outcomes is \( 8^5 \).

**Calculation:**
\[ 8^5 = 32768 \]

### Step 2: Determine the Favorable Outcomes
Next, we need to find the number of favorable outcomes where all 5 passengers alight at different floors. This can be done by selecting 5 floors out of the 8 and then arranging the 5 passengers on those selected floors.

1. **Choose 5 floors from 8:** The number of ways to choose 5 floors from 8 is given by the combination formula \( \binom{n}{r} \):
\[ \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

2. **Arrange 5 passengers on the 5 chosen floors:** The number of ways to arrange 5 passengers on 5 floors is given by \( 5! \):
\[ 5! = 120 \]

3. **Total favorable outcomes:** The total number of favorable outcomes is the product of the number of ways to choose the floors and the number of arrangements of passengers:
\[ \text{Favorable outcomes} = \binom{8}{5} \times 5! = 56 \times 120 = 6720 \]

### Step 3: Calculate the Probability
The probability \( P \) that all 5 passengers alight at different floors is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{6720}{32768} \]

### Step 4: Simplify the Probability
To simplify \( \frac{6720}{32768} \):
- Divide both the numerator and the denominator by their greatest common divisor (GCD).
- The GCD of 6720 and 32768 is 16.

**Simplification:**
\[ \frac{6720 \div 16}{32768 \div 16} = \frac{420}{2048} \]

### Final Answer:
The probability that all 5 passengers alight at different floors is:
\[ \frac{420}{2048} \approx 0.2041 \]