Please show your work using MDM4U (Grade 12 Data Management) methods. Thank you so much for your help, i really need it. Please no ai or chatgpt, I need to knoow how to do these for future tests:

5. There are 10 men, 14 women, and 16 children on a harbour cruise boat. Seven passengers are selected at random to win special prizes.

a) What is the probability that the winners are all children? [1]

b) What is the probability that exactly two are children? [2]

c) What is the probability that exactly three are children and two are women? [2]

d) What is the probability that at most two men are selected? [3]

e) What is the expected number of winners are i) men? ii) women? iii) children? [2]