Can I address some of the things that myprivateidaho| said:
"I know for example that it's fairly rare for state school kids to be able to study maths at Cambridge"
I found the 2014 Cambridge university admissions statistics. In 2014 there were 114 State school acceptances and 32 Independent acceptances to read maths.
Maths is an odd one - there are far more offers for maths as these will include STEP exams, which are much harder than a-levels. For most other subjects if you get an offer at interview you will almost certainly get your place (assuming most people applying to Cambridge will be in course for A*'s). On the positive side, a state school candidate who has not had "interview coaching" will be more likely to get an offer in maths, but very few state schools have teachers who can support students in STEP.
"most state schools just don't teach enough maths for them to do the course well and many only have provision for students to do one maths a-level"
The Further Maths support network was set up to help all schools offer at least AS Further maths. However, Independent schools still have some massive advantages. They tend to teach the A-level maths syllabus in one year, then the further maths syllabus in the next whereas in state schools you tend to have A-level lessons with one class and then some extra lessons on top for some further maths modules - this tends to mean that they end up doing Decision Maths which is of no use at all for Cambridge - really you need as much Pure and Mechanics as you can (Statistics is somewhat useful, but not as much as the others). Having a separate group who are the further mathematicians means that they can race through the easier modules (C1/C2 etc) and spend more time on the harder ones - independent students are also more likely to do both FP2 and FP3.
IMO, three A-levels in maths is not as useful as 2 good ones (with lots of Pure and Mechanics) with further reading on top.
Then there is STEP. Most kids who want to study maths at cambridge have found maths easy (at GCSE etc). Then they sit down and are faced with a paper (usually 2) where they don't know what to do - most STEP questions require some thought and ingenuity, and candidates have to do 6 questions in 3 hours (rather than the 9/10 in 1.5 hours for a maths module). STEP questions also link lots of different areas together. It is very important that candidates get used to "being stuck" a long time before they sit the exams!
There is some stuff out there for prospective maths students, which can be found at maths.org/step/ . They are designed to try and increase students confidence with this sort of question, and there is a forum where they can ask for help. The assignments are pretty good, and there is a link to Siklos's book (freely available as a pdf).