A person writes letters to six friends and addresses the corresponding envelopes. Let x be the number of ways so that atleast two of the letters are in wrong envelopes and y be the number of ways so that all the letters are in wrong envelopes. Then, x-y is equal to

Options

A. 719
B. 265
C. 454
D. None of these

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) and then compute \( x - y \).

### Step 1: Calculate \( x \)
**Definition of \( x \)**: \( x \) is the number of ways such that at least two of the letters are in wrong envelopes.

To find \( x \), we can use the principle of inclusion-exclusion. We first calculate the total arrangements of letters and then subtract the cases where fewer than two letters are in wrong envelopes.

1. **Total arrangements**: The total number of ways to arrange 6 letters in 6 envelopes is \( 6! = 720 \).

2. **Cases to subtract**:
- **Case 1**: Exactly 0 letters in wrong envelopes (all correct): There is only 1 way (the correct arrangement).
- **Case 2**: Exactly 1 letter in the wrong envelope: Choose 1 letter to be in the wrong envelope and the remaining 5 must be in the correct envelopes. This is impossible since if one letter is wrong, at least one other must also be wrong. Thus, this case contributes 0 ways.

Using inclusion-exclusion:
\[ x = \text{Total arrangements} - \text{(0 letters wrong)} - \text{(1 letter wrong)} \]
\[ x = 720 - 1 - 0 = 719 \]

### Step 2: Calculate \( y \)
**Definition of \( y \)**: \( y \) is the number of ways such that all letters are in wrong envelopes.

This is known as a derangement, denoted as \( !n \). The formula for derangements is:
\[ !n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \]
For \( n = 6 \):
\[ !6 = 6! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \frac{1}{6!} \right) \]
Calculating this:
\[ !6 = 720 \left( 1 - 1 + 0.5 - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} + \frac{1}{720} \right) \]
Calculating the series:
\[ = 720 \left( 0 + 0.5 - 0.1667 + 0.04167 - 0.00833 + 0.00139 \right) \]
\[ = 720 \left( 0.36667 \right) \approx 265 \]

### Step 3: Calculate \( x - y \)
Now, we can compute:
\[ x - y = 719 - 265 = 454 \]

### Final Answer
Thus, \( x - y = 454 \).

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