In more detail, you need to discuss why you have decided to become a solicitor instead of a barrister—and you definitely shouldn't say it's because pupillages are more competitive than training contracts.
It'll be worth talking about a specific area of law you're interested in—remember all those core modules from your LLB or GDL? If you don't have a preferred area yet, don't worry. You can talk about different aspects of a few, or even pretend to be more enthusiastic about one area than you really are. All they're looking for is evidence that you're interested and knowledgeable.
The contents of the LPC don't vary much from provider to provider, so anything you say about the course itself will apply to all three of your choices. However certain law school will specialise in certain areas of law, so it's probably good to discuss these and explain your reasoning for choosing these institutions.
Mentioning a specific aspect of law and explaining that you intend to go into this area in your career also shows your provider that you're thinking seriously about your long-term plans for the future. Everyone applying for the LPC knows they want to be a lawyer, but by proving that you have a specific direction you'll set yourself apart from candidates who are just happy to take any legal career path.
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You've shown why you want to do the LPC at their institution, but now you need to show why they should want you. If you've got to this point in your legal career, chances are you'll have a lot of achievements and skills you can list. You don't need to list every hobby or prize you've ever got: your certificate for best spelling test results in Year 3 won't help you on the LPC.
Make sure everything you're writing can be related to your future career as a lawyer. That doesn't mean you can only write about any legal work experience or your involvement with the law society at university, though you should definitely mention this at length. If you were the president of Chess society, you can explain how this improved your leadership and team-management skills.
You should also mention any academic achievements you may have; an award for being the best student in your year would be ideal, but you could also mention any particularly impressive exam results—as long as they relate to an area of law you're interested in.
Although you have 10,000 characters, you don't have to use them all. It's better to be concise than ramble on, and succinctness is a valuable skill for lawyers to have.
It's best to write your statement out in full sentences and don't use bullet points. It should go without saying that your grammar, spelling and punctuation need to be perfect, so get a friend or family member to proofread it.