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In my post, I will solve only one part, namely part (a).
(a) "equal doctors and nurses" means "3 doctors and 3 nurses".
The answer is P(3 of 5; 3 of 10) = = = 0.23976 (rounded). ANSWER
Solved.
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I do not like to solve many problems in one post, because I think that it is BAD WAY to teach.
By the way, the rules of this forum require ONE problem per post. See the page
https://www.algebra.com/tutors/students/ask.mpl?action=ask_question&topic=Equations&return_url=http://www.algebra.com/algebra/homework/equations/
from which you post your problems.