Washing Detergent Probability Riddle

A washing detergent company is running a new promotion. Each box of washing detergent contains exactly one coupon randomly selected (with equal probability) from six different types of coupons. To claim a prize, the washing detergent company requires a customer to present a complete set containing one of each of the different types of coupons. How many boxes are required on average to claim a prize?

Discuss your solution. Can you still solve the problem if the number of different types of coupons was increased to ten (and you still need to collect a complete set)?