2023 WSMO Accuracy Round Problems/Problem 4

Problem

Bob and his 3 friends are standing in a line of 10 people. Given that Bob is not on either end of the line, then the probability the person in front and behind Bob are both his friends is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

The probability the person in front of Bob is his friend is $\frac{3}{9}$ . The probability that the person behind Bob is his friend given that the persin in front of Bob is his friend $\frac{2}{8}$ . So, our answer is $\left(\frac{3}{9}\right)\left(\frac28\right)=\frac6{72}=\frac1{12}\implies1+12 = \boxed{13}.$

~pinkpig

Page

Toolbox

Search

2023 WSMO Accuracy Round Problems/Problem 4

Problem

Solution