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? ( )

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- Dec 15th 2011, 07:02 AMgodelproofRolling dice probability
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? ( )

- Dec 15th 2011, 08:22 AMgodelproofRe: Rolling dice probability
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 .