A clerk is given 3 letters written in a language unknown to her. She is also given an envelope with an address

Question

A clerk is given 3 letters written in a language unknown to her. She is also given an envelope with an address corresponding to each letter written in the same language. If the clerk puts the letters in the envelopes, in how many ways can she do it so that none of the 3 envelopes contains the letter

Accepted Answer

This problem asks for the number of complete derangements of

n = 3

items, where no item is placed in its original correct spot.
The formula for the number of derangements (

! n

) is given by:

! n = n! (1 - \frac{1}{1 !} + \frac{1}{2 !} - \frac{1}{3 !} + \dots + \frac{( - 1 ) ^{n}}{n !})

Substituting

n = 3

:

! 3 = 3! (1 - 1 + \frac{1}{2} - \frac{1}{6})

! 3 = 6 (\frac{1}{2} - \frac{1}{6}) = 6 (\frac{3 - 1}{6}) = 6 (\frac{2}{6}) = 2

Thus, there are exactly

2

ways to arrange the letters such that every single letter goes into a completely wrong envelope.

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