If n letters are placed into n addressed envelopes at random, the probability that at least one letter will go into a wrongly addressed envelope is ____.

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Detailed Solution

Generally, we find probability using the formula,
P(E)=n(F)n(E)
Here P (E) represents the probability of the event, n (F) represents a total number of favorable outcomes of the event and n (E) represents a total number of outcomes for an event.
Now, we know that, the total number of permutations or ways of placing letters are n!
Also, the number of ways in which all letters are placed in the correct envelope is 1.
So for at least 1 letter to be misplaced to compliment the set which can be done in n! - 1 way.
Hence, we know the total number of events or n (E) are n! and the number of favorable outcome or n(F) are (n!-1) so on substituting we get, probability or P(E) as,
P(E)=n(F)n(E)⇒(n!−1)/n!⇒1−1/n!.

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