Final answer:
The question involves calculating the probability of selecting students from different grades in a marching band to lead practice using combinatorial methods.
Step-by-step explanation:
The question deals with finding the probability of certain combinations of students from different grade levels in a marching band being chosen for specific roles. Given there are 4 freshman, 7 sophomores, 6 juniors, and 3 seniors clarinet players, and two are chosen at random, we need to calculate different probabilities related to their grade levels.
Neither is a senior: The probability that neither of the chosen students is a senior can be found by subtracting the total number of ways to choose two non-seniors from the total number of ways to choose any two students.
One is a freshman and the other is a sophomore: This probability can be calculated by finding the number of ways one freshman and one sophomore can be chosen and then dividing by the total number of ways to choose any two students.
At least one is a sophomore: The probability for this is found by calculating one minus the probability that no sophomores are chosen.
Both are juniors: The probability of both chosen students being juniors is calculated using the combination formula.
Each calculation uses the principles of combinations to find the probabilities.