A relay team of 3 is chosen at random from 7 athletes, two of whom are brothers. What is the probability that EXACTLY one of the two brothers is in the team?

Teams number $C(7,3)=35$. For exactly one brother, pick which brother is in — 2 ways — and fill the other two places from the 5 athletes who are not brothers, $C(5,2)=10$ ways. That gives $2\times 10=20$ teams and a probability of 20/35 = 4/7. The value 1/7 = 5/35 covers BOTH brothers being chosen, from $C(5,1)=5$, since the third member is then any one of the other five. The value 2/7 = 10/35 covers NEITHER brother, from $C(5,3)=10$. The value 5/7 = 25/35 is 'at least one brother', which lumps the exactly-one and both cases together and so overshoots. The three cases partition the teams, and as a check 1/7 + 4/7 + 2/7 = 1.