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Combinations and Permutations Problem Set

Question 1:
From a total of 15 people, three committees consisting of 3, 4, and 5 people, respectively, are to be chosen.

b) How many such sets of committees are possible if there is no restriction on the number of committees on which a person may serve?

Question 2:
c) In how many ways can we select a set of five microprocessors containing exactly two defective microprocessors?

d) In how many ways can we select a set of five microprocessors containing at least one defective microprocessor?

Question 3:
In how many ways can 15 identical computer science books and 10 identical mathematics books be distributed among five students?

Question 4:
A major software company is arranging a job fair with the intention of hiring 10 recent graduates. No candidate can receive more than one offer. In response to the company's invitation, 150 candidates have appeared at the fair.
c) How many ways are there to extend job offers to 10 of the 150 candidates, if we already know that at least one of Ron Smart and Fran Wise gets an offer?

d) How many ways are there to distribute the 150 resumes to three interviewers?

e) How many ways are there for three interviewers to select three resumes (one resume for each interviewer) from the pile of 150 resumes for the first interview round?

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Solution Preview

Question 1:
b) Since a person may serve multiple committees,
For the first committee of 3 people, there are C(15,3) ways;
For the second committee of 4 people, there are still C(15,4) ways rather than C(12,4) since a person can serve multiple committees;
For the third committee of 5 people, there are still C(15,5) ways.
Since these three committees are independent, the total ways: C(15,3)C(15,4)C(15,5).

Question 2:
c) 95!/(92!*3!) * 5!/(3!*2!)
It is 95!/(92!*3!)*5!/(3!*2!)=95*94*93/(3*2*1)*(5*4)/(2*1)= 1384150

d) This is a complement event. Put in another way, we first figure out the total ways of selecting 5 microprocessors and in ...

Solution Summary

The solution discusses the combinations and permutations problem set.