Question

If 5 letters are to be placed in 5-addressed envelopes, then the probability that at least one letter is placed in the wrongly addressed envelope is:

• A. \( \frac{1}{5} \)
• B. \( \frac{1}{120} \)
• C. \( \frac{4}{5} \)
• D. \( \frac{119}{120} \)

Hint:

Use complementary counting for probability questions involving derangements (incorrect placements).

Answer 1

The Correct Option is D

Step 1: Finding Probability of Correct Placement
Total number of ways to place 5 letters into 5 envelopes: \[ 5! = 120 \] Only 1 way exists where all letters are correctly placed. Step 2: Using Complementary Probability
The probability that all letters are correctly placed: \[ P(\text{all correct}) = \frac{1}{5!} = \frac{1}{120} \] The probability that at least one letter is wrongly placed: \[ P(\text{at least one wrong}) = 1 - P(\text{all correct}) \] \[ = 1 - \frac{1}{120} = \frac{119}{120} \] Thus, the correct answer is \( \frac{119}{120} \).