19. There are eight different sizes of puzzles. Each person selects a puzzle and keeps it. What is the probability that the first puzzle chosen is a 100 -piece puzzle and the second is a \( 1000- \) piece puzzle?

To determine the probability that the first puzzle chosen is a 100-piece puzzle and the second is a 1000-piece puzzle, we start by noting that there are eight different sizes of puzzles. Assuming each size has the same probability of being chosen: The probability of choosing a 100-piece puzzle first is \( \frac{1}{8} \). After one puzzle is chosen, there are still seven options remaining, so the probability of choosing a 1000-piece puzzle second is also \( \frac{1}{8} \). To find the combined probability of both events happening in sequence, we multiply the probabilities: \[ P(\text{100-piece and then 1000-piece}) = P(\text{100-piece}) \times P(\text{1000-piece}) = \frac{1}{8} \times \frac{1}{8} = \frac{1}{64}. \] So, the probability that the first puzzle chosen is a 100-piece puzzle and the second is a 1000-piece puzzle is \( \frac{1}{64} \).