Name:      

Directions: Show all work. You may leave your answer in terms of factorials, falling factorials, and binomial coefficients. Any sums or products should be as simple as possible.

1. [5 parts, 2 points each] A fair $6$-sided die is rolled $3$ times.

   1. Give the sample space $\mathrm{\Sigma }$.
   2. What is the probability that all $3$ rolls give the same value?
   3. What is the probability that all $3$ rolls give distinct values?
   4. What is the probability that among the $3$ rolls, one of the values is distinct and the other two are equal?
   5. What is the probability that at least $1$ of the rolls is a $4$?