Cut the Cards

[David Hayden]

PPC Journal V5N4 Page 13 poses a question about a deck of cards. Choose a card at random from a deck of 52 and put the card back. What is the average number of choices required to see all 52 cards?

I simulated the experiment in C++ and got the right answer (236) but I can't figure how to do this analytically. Is it possible?

If you simplify the problem down to a deck of 2 cards then it's easier: you choose the first card, and then it's just a matter of the average number of choices to get the second: 1*1/2 + 2*1/4 + 3*1/8 ..., but can that be extended to more cards?

Dave