a) To calculate the probability that all five awards are given to students majoring in mathematics, we need to find the probability of selecting a mathematics major for each award and multiply them together. The first award can be given to any of the 15 mathematics majors out of the total 28 applicants. So the probability of selecting a mathematics major for the first award is 15/28. For the second award, one mathematics major has already received an award, leaving 14 mathematics majors out of the remaining 27 applicants. Therefore, the probability of selecting a mathematics major for the second award is 14/27. Similarly, for the third, fourth, and fifth awards, the probabilities are 13/26, 12/25, and 11/24, respectively. To find the probability that all five awards are given to students majoring in mathematics, we multiply these probabilities together: (15/28) * (14/27) * (13/26) * (12/25) * (11/24) ≈ 0.018 or 1.8% Therefore, the probability that all five awards are given to students majoring in mathematics is approximately 0.018 or 1.8%. b) To find the probability that none of the five recipients are majoring in mathematics, we need to find the probability of selecting a non-mathematics major for each award and multiply them together. The first award can be given to any of the 13 non-mathematics majors out of the total 28 applicants. So the probability of selecting a non-mathematics major for the first award is 13/28. For the second award, one non-mathematics major has already received an award, leaving 12 non-mathematics majors out of the remaining 27 applicants. Therefore, the probability of selecting a non-mathematics major for the second award is 12/27. Similarly, for the third, fourth, and fifth awards, the probabilities are 11/26, 10/25, and 9/24, respectively. To find the probability that none of the five recipients are majoring in mathematics, we multiply these probabilities together: (13/28) * (12/27) * (11/26) * (10/25) * (9/24) ≈ 0.013or 1.3% Therefore, the probability that none of the five recipients are majoring in mathematics is approximately 0.013 or 1.3% ** Check the calculation to be sure