Learn / Library / Combinatorics

Letters in wrong envelopes

CombinatoricsEasy

A clerk puts $3$ different letters into $3$ pre-addressed envelopes, one letter per envelope, completely at random, with all $6$ arrangements equally likely. What is the probability that no letter lands in its correct envelope?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

List the arrangements of $3$ items where nothing stays in place.

Solution

The only arrangements of $3$ items with no fixed point are the $2$ cyclic shifts.
Out of $6$ equally likely arrangements, $2$ are good.
The probability is $\frac{2}{6} = \frac{1}{3}$ .
The answer is $\frac{1}{3}$ .

Answer: $\frac{1}{3}$

Next question Back to the library