Derangements

Question

The number of ways in which 4 letters can be placed in 4 addressed envelopes such that no letter goes to the correct envelope is

RankGuru · Accepted Answer

1. **Problem Identification**: This is a direct application of the 'Derangement' formula, $D_n$. 
2. **Formula**: $D_n = n! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(-1)^n}{n!} \right)$.
3. **Substitute $n=4$**:
   - $D_4 = 4! \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \right)$
4. **Calculation**:
   - $D_4 = 24 \left( \frac{12 - 4 + 1}{24} \right)$
   - $D_4 = 12 - 4 + 1 = 9$.
5. **Result**: 9 ways.

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