In one of the comments you ask how to approximate this when this is given as an interview question and you don't have your calculator. Well, the birthday problem is well-known, and it is indeed well known that with about 23 people the chance of there being two people sharing a birthday is about $50$%. So, with 25 people here should be a chance of a little over $50$% that two or more people share a birthday (meaning the expectednumber of hands is defintely above $1$ ... though the probability of there being three or more is most likely still fairly low. So, eyeballing that, you'd end up with something above $1$ though still well below $2$.

In one of the comments you ask how to approximate this when this is given as an interview question and you don't have your calculator. Well, the birthday problem is well-known, and it is indeed well known that with 23 people the chance of there being two people sharing a birthday is about $50$%. So, with 25 people here should be a chance of a little over $50$% that two or more people share a birthday, meaning that the expected number of hands is defintely above $1$. On the other hand, the probability of there being three or more is a good bit lower. So, eyeballing that, you'd end up with something above $1$ though still well below $2$. Personally I would have guessed in the $1.2$ or $1.3$ neighborhood, so I am a bit surprised it's close to $1.6$, but I bet my answer would've satisfied the interviewer. :)