I also thought I'd give you a few pieces of advice about the application process and personal statements.
1) Only apply for one course - you can apply to more than one course, but you can only be accepted to one. They will look at your applications for two or more courses, decide which one your personal statement is best suited to, and reject you from the other courses instantly. So it doesn't increase your chances of getting an offer in any way - in fact, it's a bit of a waste of one of your 5 choices. Furthermore, LSE really like it when you tailor your personal statement directly to their course, and if you have tried to make your PS relevant to two different courses, it may end up being not good/similar enough for either of them.
2) Personal statements - quality not quantity, analysis not listing - A personal statement is not a list of all your achievements, extra-curriculars and every book you've ever read. Don't namedrop a book you read or a talk you went just for the sake of mentioning it - instead, find an aspect or fact from it that you find really interesting, and talk about it. Try to draw connections with your other reading/extra-curriculars. Where possible, show how one talk sparked your interest in - e.g. decolonisation/American politics/global financial markets - and so you developed that interest by reading more about it, or listening to a podcast/watching a documentary about it. Show your learning process, and the way your mind draws connections between things.
3) Explain WHY something interested you - DON'T say "attending maths sessions after school furthered my interest in mathematics." DO say "in after-school maths sessions, I first learnt about *theory/phenomenon X* - this inspired me to explore it further by reading Author Y's book on *topic X*; what most stood out to me from this book was *fact Z*."
4) As far as possible, tailor your personal statement to the LSE course - LSE like personal statements that are written about the LSE course. They are not impressed when your personal statement talks about subjects that you have not applied to study at LSE.
For example, you may be planning on applying to Politics and International Relations at 4 unis, and Government at LSE, because it has a higher acceptance rate than LSE Politics & IR. However, as the Government course contains *no IR*, if your Personal Statement is half-focused on why you love IR, this will not please the Government academics who will be assessing your application.