Counting and probability

Decide whether order matters, then count the complement when 'at least one' appears. A probability is favorable outcomes over equally likely outcomes.

A lesson for this area

Problem 1AMC 12

A club with $9$ members chooses a president, a vice president, and a treasurer. No member may hold more than one office. In how many ways can the three offices be filled?

Problem 2AMC 12

A fair six-sided die with faces numbered $1$ through $6$ is rolled three times. What is the probability that the three numbers rolled are in strictly increasing order?

Problem 3AMC 12

Three identical red balls and three identical blue balls are placed into $4$ different boxes. Any box may hold any number of balls. In how many ways can this be done?

Problem 4AMC 12

A positive divisor of $720$ is chosen at random, with every positive divisor equally likely. What is the probability that it is a perfect square or a perfect cube?

Problem 5AMC 12

How many $5$ -element subsets of $\{1, 2, 3, \ldots, 10\}$ contain at least one pair of consecutive integers?