We would like to organize the following prize game (simplified):
We print five letters, say, $10 \times A$, $10 \times B$, $5\times C$, $1 \times D$, $50\times E$. A customer draws a letter at each purchase.
1) What is the probability that after three visits, he has $A, B$ and $C$?
2) How does this probability change if he makes not $3$, but $4, 5$ or $6$ purchases?
3) What is the probability that after $4$ ($5,6,7,\ldots$) visits he has $A,B,C,D$?
4) How many purchases he needs to make in order to achieve given probability, say for $ABC$?
Of course in reality the numbers will be higher.