As a broad assumption - Most firms who hold open days simply provide places on a first-come-first-served basis. This is almost exclusively true for US Firms (anecdotal experience here). This is absolutely not the case for firms like CC, who last year advised they had over 800 applications for their open days, for which there was cumulatively 250 spots or so. In this respect, you must tailor your application, and this is also an excellent practice for your future applications.
Not sure US firms recruit any differently for open days than UK firms. We certainly review your applications - it's not first-come-first-serve.