2024 SSMO Speed Round Problems/Problem 6
Problem
There are 
people and houses. Each person independently randomly chooses a house to live in. The expected number of inhabited houses can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Let 
equal 
if the nth house is occupied and 
if the nth house is unoccupied. Thus, we are trying to find ![$\mathbb{E}[a_1+a_2+a_3+a_4] = 4\mathbb{E}[a_1]$](http://latex.artofproblemsolving.com/2/8/b/28b16f035372bc7b85c7c31849f99abdb0d0c0ee.png)
by the Linearity of Expectation.
