Referring to Example B in Section 4.1.2, what is the expected number of coupons needed t...

Referring to Example B in Section 4.1.2, what is the expected number of coupons needed to collect r different types, where r < n?

Reference

Coupon Collection

Suppose that you collect coupons, that there are n distinct types of coupons, and that on each trial you are equally likely to get a coupon of any of the types. Howmany trials would you expect to go through until you had a complete set of coupons? (This might be a model for collecting baseball cards or for certain grocery store promotions.) The solution of this problem is greatly simplified by representing the number of trials as a sum. Let X₁ be the number of trials up to and including the trial on which the first coupon is collected: X₁ = 1. Let X₂ be the number of trials from that point up to and including the trial on which the next coupon different from the first is obtained; let X₃ be the number of trials from that point up to and including the trial on which the third distinct coupon is collected; and so on, up to X_n. Then the total number of trials, X, is the sum of the X_i , i = 1, 2, . . . , n.

For example, if there are 10 types of coupons, the expected number of trials necessary to obtain at least one of each kind is 29.3.