Interchanging

Probability Level 2

In how many ways 7 letters can be placed in 7 envelopes such that no letter is placed in its corresponding envelope?

1 solution

Alpana Singhal
Aug 28, 2014

it is a dearngemnt problem n! [ 1 - 1/1! + 1/2! - 1/3! ............ 1/n!]

for placing n letters in n wrong envelopes

Another formula to quickly evaluate the value of $!n$ is:

$!n=\left \lfloor \frac{n!}{e} \right \rceil$

where, $f(x)=!x$ is the subfactorial function used to find the number of possible derangements of $n$ items, $f(x)=\left \lfloor x \right \rceil$ is the nearest integer function , $f(x)=x!$ is the factorial function and $e$ is the Euler's number .

Prasun Biswas - 6 years, 4 months ago