Roll an -sided fair dice times. What is the probability that all numbers from to appear at least once and none from to appear during the trials? ( )
Here's my thought. Check if it is correct. I also want to know if there's a simpler method.
Given any numbers from to , let be the number of scenarios under which all and only those numbers appear in our trials. Then we have 1, and .
By this definition, there are exactly scenarios under which exactly different numbers from to appear.
Hence the required probability .