What indicated that order matters in this question?

Solution 1:

$P(G',G',G')$ is the probability all three didn't win gold. The event $\{G',G',G'\}$ is order independent; reordering three non-Gold winning statuses among three players still makes them three non-Gold winners, the same event.

Update: As suggested, the above is sloppy notation, intended for heuristic purposes. And a notion of "ordering" is not necessary. To write it in terms of Bernoulli trials, letting $X_i$ denote the random variable given by the indicator for person $i$ winning gold, then the set $(\cap_i\{\omega:X_i(\omega)=0\})^c$ is the same as $\cup_i\{\omega:X_i(\omega)=1\}$ by De Morgan's laws.

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