The question is a bit ambiguous.

After hashing of how many keys, will the probability that any new key hashed collides with an existing one exceed 0.5.

Here, 'new key hashed' is the ambiguity. It can mean the probability of a collision in the next 'hash', or the probability of a collision in any of the hashes of the 'new keys' starting from the first insertion. For the first case answer must be $10$ to get probability equal to $0.5$, and so $11$ must be the answer for probability $> 0.5$. Thus we can conclude from given choices, it is the second case.

So, we need to find $n$ such that after $n$ hashes, probability of collision (in any of the $n$ hashes) $> 0.5$.

Probability that there will be a collision after $n$ hashes (a collision happened in at least one of those $n$ hashes) $= 1 - $Probability that there was no collision in the first $n$ hashes

$= 1 - 1. \frac{19}{20} . \frac{18}{20} \dots \frac{20-n+1}{20}$.

So, we need,

$0.5 < 1 - 1. \frac{19}{20} . \frac{18}{20} \dots \frac{20-n+1}{20}$.

$\implies \frac{19}{20} . \frac{18}{20} \dots \frac{20-n+1}{20} < 0.5$.

For $n=5$, we get, $0.5814$ and for $n=6$, we get $0.43605$. So, answer should be $n = 6$.

Correct Answer: $B$