I think Rachel Riley's mental maths is reputed to be excellent. In which case she probably knows all the tricks needed to see whether or not a target is possible without having to do much calculation. This isn't that unreasonable. For example, given the numbers 2, 5 and 10, usable only once each at most, it's clear that there is no way to get above 100 (2*5*10), and most numbers above 50 would be impossible too (apart from 52 and 70). With experience, you could probably perform similar tricks for six numbers and a target of no greater than 1000.
Alternatively, it's possible that she might have gone backwards to some extent. Imagine being given a target of 773. How many ways are there to make this with six numbers? It might well be that you can prove that there's no way to do it unless you have a 7, or some way to make a 7, so a quick inspection of the numbers 2, 4, 8, 10, 25, 100 will show that (if it were true that you needed a 7) the target is impossible. Again, that is the sort of thing that might come with practice.
I have made that last example up, so if anyone can find 773 given those numbers I'd not be surprised.
In the end I suppose it just comes down to experience.