A. How many ways can a group of 7 out of the 12 members be chosen to work on a s

Question

A. How many ways can a group of 7 out of the 12 members be chosen to work on a special project? Explain.
B. Suppose that out of 12 members on the team 7 are IT majors and 5 are math majors.
(i) How many groups of 7 people can be formed that contain 4 IT majors and 3 math majors? Explain.
(ii) How many groups of 7 can be formed that contain a least 1 IT major? Explain.
C. Suppose that two team members (of the 12 of either major) refuse to work together. How many groups of 7 can be chosen to work on a project? Explain.
D. Suppose that two team members insist on either working together or not at all on projects. How many groups of 7 can be chosen to work on a project? Explain.

Explanation / Answer

a) 12*11*10*9*8*7*6 = 3991680 ways. At first, you have 12 choices to pick first member, and 11 choices to pick second member, 10 choices to pick third member, and so on. b) Given 7 IT members and 5 math majors, to choose 4 IT members, (7*6*5*4) = 840 ways to pick IT members. And to choose 3 math majors, (5*4*3) = 60 ways. Therefore, overall, there are 840*60 = 50400 ways to form such group. c) Since there are two members who refuse to work together, you can exclude them. Therefore, 10*9*8*7*6*5*4 = 604800 ways to form group of 7. d) If two members insist to work together, you can count two people as one person: therefore, there are 11*10*9*8*7*6*5 = 1663200 ways. From C) we know there are 604800 ways to form group when there are two members who refuse to work together. Therefore, add these two: 2268000 ways in total!

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