A teacher decided to encourage the kids by distributing prizes to them. Each of the prizes was different from the other. The total number of prize was $k$, and total number of kids was $p$. To encourage the kids, the number of prizes was more than the number of kids. But, the teacher imposed a restriction on herself that each kid would receive $(k – 1)$ prizes at the most. How many ways she could distribute the prizes?
Answer is $p^k-p$
If I understood the problem correctly, it is no-where stated that a student can get no prize at all, and the answer seems to be using this assumption,however I am a bit confused why the answer is not $p^{k-1}$?
If the restriction is that no student can get more than $(k-1)$ prizes then what is wrong with starting with $(k-1)$ distinct prizes and distributing those into $p$ distinct groups (students)?