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Solution: We are given 5 distinct transitions occurring at unique seconds in a 3600-second interval, with all orderings of transition times uniformly random. We are to compute the probability that two specific transitions, say $T_1$ and $T_2$, occur at consecutive seconds in the sequence — that is, one immediately follows the other (either $T_1$ then $T_2$, or $T_2$ then $T_1$).
First, the total number of ways to assign 5 distinct transition times to 3600 seconds such that they are at distinct seconds is equivalent to choosing 5 distinct time points and permuting them in order:
But since we are computing a probability over uniformly random orderings of the 5 transitions (with fixed distinct timing positions), we can instead consider the $5!$ possible orderings of the transitions, with each ordering equally likely under the uniform model.

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