When seven basketball players are about to have a free-throw competition, they often draw names out of a hat to randomly select the order in which they shoot. What is the probability that they shoot free throws in alphabetical order? Assume each player has a different name. P(shoot free throws in alphabetical order) = (Type an integer or a simplified fraction.)

To find the probability that the basketball players shoot free throws in alphabetical order, we need to consider the total number of possible orderings and the number of favorable outcomes (i.e., the players shooting in alphabetical order). Given that there are 7 basketball players with different names, the total number of possible orderings of the players is 7! (7 factorial), which is the number of ways to arrange 7 distinct items. The number of favorable outcomes, where the players shoot in alphabetical order, is 1, as there is only one specific order that satisfies this condition. Therefore, the probability of the players shooting free throws in alphabetical order is given by: \[ P(\text{shoot free throws in alphabetical order}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible orderings}} = \frac{1}{7!} \] Let's calculate this probability. Simplify the expression by following steps: - step0: Solution: \(\frac{1}{7!}\) - step1: Calculate the value: \(\frac{1}{5040}\) The probability that the basketball players shoot free throws in alphabetical order is \( \frac{1}{5040} \).