Please help me this probability question

Question

A grocery store is sponsoring a sale promotion where the cashier give away one of letters A, E ,L,S ,U ,V for each purchase. If a customer collects all six letters he gets 10 dolar worth for grociers free.What is the expected number of trips to the store a customers need to make in order to get a complain set? Assume differen letters are given away randomly.

Answer 1

After the first visit, there are 5 of 6 letters that have not been selected, so the expected number of visits until the second letter has been found is 6/5 (expected value with the geometric distribution if you like); after two letters have been found, 4 of 6 letters have not been selected, so the expected number of trips until the third letter is found is 6/4; this pattern continues for the next three; thus, the expected number of visits to complete a set is

1 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14 7/10