There are six letters `L_(1),L_(2),L_(3),L_(4),L_(5),L_(6)` and their corresponding six envelops `E_(1),E_(2),E_(3),E_(4),E_(5),E_(6)`. Letters having odd value can be put into odd valued envelopes and even valued letters can be put into even valued envelopes, so that no letter goes into the right envelopes, then number of arrangements equals.

Options

A. 6
B. 9
C. 44
D. 4

Answer

To solve the problem of arranging six letters \( L_1, L_2, L_3, L_4, L_5, L_6 \) into their corresponding envelopes \( E_1, E_2, E_3, E_4, E_5, E_6 \) under the given constraints, we can follow these steps:

### Step 1: Identify the Odd and Even Letters and Envelopes
- Odd letters: \( L_1, L_3, L_5 \) can go into odd envelopes \( E_1, E_3, E_5 \).
- Even letters: \( L_2, L_4, L_6 \) can go into even envelopes \( E_2, E_4, E_6 \).

### Step 2: Determine the Derangement Condition
We need to ensure that no letter goes into its corresponding envelope. This means we are looking for derangements of the letters within their respective groups (odd and even).

### Step 3: Calculate Derangements for Odd Letters
For the odd letters \( L_1, L_3, L_5 \):
- The total arrangements of 3 letters is \( 3! = 6 \).
- The derangements \( !n \) for \( n = 3 \) can be calculated as:
\[ !3 = 3! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} \right) = 6 \left( 1 - 1 + 0.5 - \frac{1}{6} \right) = 6 \left( 0.5 - \frac{1}{6} \right) = 6 \left( \frac{3}{6} - \frac{1}{6} \right) = 6 \left( \frac{2}{6} \right) = 2. \]
Thus, there are \( !3 = 2 \) derangements for the odd letters.

### Step 4: Calculate Derangements for Even Letters
For the even letters \( L_2, L_4, L_6 \):
- Similarly, the derangements \( !3 \) for these letters will also be:
\[ !3 = 2. \]

### Step 5: Combine the Derangements
Since the arrangements of odd and even letters are independent, the total number of arrangements where no letter goes into the correct envelope is:
\[ \text{Total arrangements} = !3 \times !3 = 2 \times 2 = 4. \]

### Final Answer
The number of arrangements such that no letter goes into the right envelope is \( \boxed{4} \).

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