How many banks should I apply to?
You should apply to as many banks as possible to increase your chances to get an interview and eventually an offer.
Generally students apply to as many as 20 investment banks for the summer internship program where there are many spots available.
For example, in London the biggest banks (e.g. Morgan Stanley, JP Morgan, Citigroup etc.) tend to recruit 60-70 summer analysts per year and on average banks receive 10,000 CVs for the summer internship program.
Boutique banks (Armapartners, Moelis & Company, Evercore etc.) generally recruit 5-10 summer analysts per year.