Let's get all the prime factors of 2450

2450 = 2×5×5×7×7

We need to make three groups from these prime factors.  The factor 1
won't help us because 1-year old babies aren't fans.

The oldest the youngest could be is 7 years old. For if the youngest 
were the next possible age, 2×5 = 10 years, the product of the other
two ages would have to be 5×7×7, and any way you group those factors,
one of the others would have to be 7 or 5, both younger than 10.
So there is no doubt that the oldest age the youngest can be is 7.

The other two could be another 7 year-old and a 
2×5×5 = 50-year old.

1.  That's one way, ages 7, 7 and 50 because 7×7×50 = 2450

The other two could be a 2×5 = 10-year-old, and a 5×7 = 35 year old:

2.  That's a second way, ages 7, 10, and 35, because 7×10×35 = 2450.

Or the other two could be 2×7 - 14-year-old. and a 5×5 = 25 year-old.

3.  That's a third way, ages 7, 14, and 25, because 7×14×25 = 2450

Those three are tho only ways, unless you would allow 1-year-old babies,
2-year old toddlers, or 5-year old kindergarteners.  I think they'd all
have to be at least in elementary school to be a fan, don't you think?  

Edwin