# 2021 JMPSC Invitationals Problems/Problem 6

## Problem

Five friends decide to meet together for a party. However, they did not plan the party well, and at noon, every friend leaves their own house and travels to one of the other four friends' houses, chosen uniformly at random. The probability that every friend sees another friend in the house they chose can be expressed in the form $\frac{m}{n}$. If $m$ and $n$ are relatively prime positive integers, find $m+n$.

## Solution

There are $4^5$ ways for the friends to choose houses. Now, we have friends can meet in pairs, so $(2,3)$ is the only way they can meet. There are $20$ ways for them to choose the $2$ houses to meet, and $3$ ways to rotate the "extra" partner (2 houses are already selected) The answer is $$\frac{2^2 \cdot 3 \cdot 5}{2^{10}} \implies \frac{15}{256} \implies 256+15=\boxed{271}.$$ ~Geometry285

## Solution 2

$$\frac{2^2 \cdot 3 \cdot 5}{4^{5}} \implies \frac{15}{256}$$Therefore, $15+256=\boxed{271}$

- kante314 -