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Which GOAT is more likely to win?

The question is pretty simple. The board has the first $X$ players and you have the other ones. All of them win if at least one player becomes the GOAT.
What is the probability that at least one of the players will be the GOAT?
Let $n$ be the number of players. Let $A_j$ be the event that the j-th player becomes the GOAT.
Which one of the following cases is more likely?
Case 1: All players are of same strength, so the probability that all of them fail is $1$.
Case 2: Each player wins the tournament with probability $p$ independently.

A:

We work with the first scenario, as it is likely the easiest.
As you said, there are $n$ players and the probability of not being the leader is $1-p$, where $p$ is some fixed probability.
The number of ways to arrange the outcomes of the $n$ players, given that none will win is $\binom{n}{k}$, where $k$ is the number of players who do not win, so the total number of possibilities is $\binom{n}{k}p^k(1-p)^{n-k}$.
As we wish all $n$ players to win, we are interested in $k=0$ and $k=n$. For a given $
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