Ravi writes 5 letters and addresses 5 envelopes. If the letters are placed in the envelopes at random, then the probability that a least one letter goes into the wrong envelope is

Options

A. `1/120`
B. `119/120`
C. `1/60`
D. `115/120`

Answer

To find the probability that at least one letter goes into the wrong envelope when Ravi writes 5 letters and addresses 5 envelopes, we can follow these steps:

### Step 1: Determine the total number of arrangements
The total number of ways to arrange 5 letters in 5 envelopes is given by the number of permutations of 5 items, which is calculated as \(5!\) (5 factorial).

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 2: Determine the number of arrangements where all letters are in the correct envelopes
There is only 1 way to place all letters in their correct envelopes. This means that the number of arrangements where all letters are in the correct envelope is \(1\).

### Step 3: Calculate the number of arrangements where at least one letter is in the wrong envelope
To find the number of arrangements where at least one letter is in the wrong envelope, we can use the complement principle. We subtract the number of arrangements where all letters are correct from the total arrangements:

\[ \text{Number of arrangements with at least one wrong} = 5! - 1 = 120 - 1 = 119 \]

### Step 4: Calculate the probability
The probability that at least one letter goes into the wrong envelope is given by the ratio of the number of arrangements with at least one wrong letter to the total number of arrangements:

\[ P(\text{at least one wrong}) = \frac{\text{Number of arrangements with at least one wrong}}{\text{Total arrangements}} = \frac{119}{120} \]

### Final Answer
Thus, the probability that at least one letter goes into the wrong envelope is:

\[ \frac{119}{120} \]

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