A person writes letters to six friends and be the number of ways so that at least two of the number of ways so that all the letters are in wrong envelopes. Then x−y=

Options

A. A. 200
B. B. 1001
C. C. 265
D. D. 454

Answer

To solve the problem of finding the number of ways to distribute letters to six friends such that at least two letters are in the wrong envelopes, we can break it down step by step.

### Step 1: Understand the Problem
We need to find the number of arrangements where at least two letters are in the wrong envelopes. This can be expressed as \( x - y \), where:
- \( x \) is the total number of arrangements with at least 2 letters wrong.
- \( y \) is the number of arrangements where all 6 letters are in the wrong envelopes.

### Step 2: Calculate Total Arrangements
The total number of ways to arrange 6 letters is \( 6! \) (factorial of 6).
\[ 6! = 720 \]

### Step 3: Calculate \( y \) (All Wrong Arrangements)
To find \( y \), we need to calculate the number of derangements (arrangements where no letter is in the correct envelope) for 6 letters. The formula for the number of derangements \( !n \) 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:
\[ = 720 \left( 0 + 0.5 - 0.1667 + 0.0417 - 0.0083 + 0.0014 \right) \]
\[ = 720 \left( 0.3681 \right) \approx 265 \]

### Step 4: Calculate \( x \) (At Least 2 Wrong)
To find \( x \), we can use the principle of inclusion-exclusion:
\[ x = 6! - \text{(arrangements with 0 wrong)} - \text{(arrangements with 1 wrong)} \]
- Arrangements with 0 wrong = 1 (all letters in correct envelopes).
- Arrangements with 1 wrong: Choose 1 letter to be wrong (6 ways), and the remaining 5 must be correct. This is not possible, so it contributes 0.

Thus:
\[ x = 720 - 1 - 0 = 719 \]

### Step 5: Calculate \( x - y \)
Now we can find \( x - y \):
\[ x - y = 719 - 265 = 454 \]

### Final Answer
Thus, the value of \( x - y \) is:
\[ \boxed{454} \]