Retail allotment odds

A main-board IPO reserves 10,50,000 shares for retail investors, with a lot of 45 shares. 4,20,000 valid retail applications are received. (a) How many applicants receive shares, and with what probability is a given application allotted? (b) A family applies with three PANs, one lot each. Find the exact probability that at least one is allotted. (c) If the average application was for 3.2 lots, what retail subscription multiple did the exchange report? Check Corollary 0.4. (d) What is the expected number of shares allotted per application?