Question

A person writes four letters and addresses four envelopes. If the letters are placed in the envelopes at random, then the probability that not all letters are placed in the right envelope is

• A. \( \frac{15}{24} \)
• B. \( \frac{11}{24} \)
• C. \( \frac{23}{24} \)
• D. \( \frac{1}{24} \)

Hint:

For problems asking “not all correct”, always consider the complement (all correct) and subtract from 1.

Answer 1

The Correct Option is C

Step 1: Identify total possible arrangements.
There are 4 letters and 4 envelopes. The number of ways to place letters in envelopes is:
\[ 4! = 24. \]

Step 2: Identify the required probability.
We need the probability that not all letters are placed correctly.
This is equal to:
\[ 1 - P(\text{all letters correctly placed}). \]

Step 3: Find probability of all correct placement.
There is only one arrangement in which all letters go into the correct envelopes.
So,
\[ P(\text{all correct}) = \frac{1}{24}. \]

Step 4: Use complement rule.
\[ P(\text{not all correct}) = 1 - \frac{1}{24}. \]

Step 5: Simplify the expression.
\[ P = \frac{23}{24}. \]

Step 6: Interpretation.
This means almost all arrangements are incorrect, and only one arrangement is perfectly correct.

Step 7: Final conclusion.
Thus, the required probability is \( \frac{23}{24} \).
Final Answer:
\[ \boxed{\frac{23}{24}}. \]