well... i guess my interpretation is not inconsistent with original problem statement, if 1-on-1 is interpreted as a one-to-one mapping i.e. a pair.
i suppose i should spend more time reading the prompt then, i guess i was solving the wrong problem
"Assume all encounters are 1-on-1 and random."
i can definitely see how one would see this and assume that in any "round" only 2 people are selected from the population, and said to have "met".
oh well, twas a good exercise of thought.