To simplify the problem, let us consider, say, only n customers, with n<10000, and only 2-digit PINs 00-99.

Is there a generalised answer to this kind of problem?

CDA (from United Kingdom)

It's not actually a simplification.

>Is there a generalised answer to this kind of problem?

There is a generalized question that can be asked:

An experiment may result in one of k equiprobable events. What is a probability that an event will not show up in n trials?

In your original problem k = 10000. To really simplify the problem, take k = 2. As a result of a trial you may have, say, either x or y. n trials produce a sequence of x's and y's. The question now is equivalent to asking how many of xy-strings of length n contain only 1 letter?

If a complete set has K coupons, how many coupon to buy (on average) until a complete N-set is collected ?

The answer is K*H(K) = K*(1/1+1/2+1/3+..+1/K)

Note: H(K) = 1/1+1/2+1/3+..+1/K) is called the Kth Harmonic number.

If 2-digit, then K=100, then
H(K)=5.18737751763.., K*H(K)=518.737751763..

If 4-digit, then K=10000, then
H(K)=9.78760603604.., K*H(K)=97876.0603604..

This does not exactly answer your question, but relates to it.

My reference are:
--Problems and Snapshots from the World of Probability, p.85
by Gunnar Blom
-- Generatingfunctionology, 4.2 p.132 in 1st edition
by Herbert S. Wilf