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 10

Five different books are placed in a row on a shelf. Two particular books must be at the two ends of the row. In how many ways can the books be arranged?

Problem 2AMC 10

A fair coin is flipped three times, and a fair six-sided die with faces numbered $1$ through $6$ is rolled once. What is the probability of getting exactly two heads and a prime number on the die?

Problem 3AMC 10

In how many ways can $10$ identical candies be given to $4$ children, named Ana, Ben, Cal, and Dee, so that each child receives at least one candy?

Problem 4AMC 10

Two different integers are chosen at random from $\{1, 2, 3, \ldots, 10\}$ , with every pair of different integers equally likely. What is the probability that their sum is even?

Problem 5AMC 10

How many five-digit positive integers have digits whose product is $12$ ?