SOLUTION: There are 12 men under consideration for the 5-man Board of Directors of the newly formed Crunch Candy Corporation. Three of these men are known to be sworn mutual en

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Question 1145543: There are 12 men under consideration for the 5-man Board of Directors of the newly formed Crunch Candy Corporation. Three of these men are known to be sworn mutual enemies and if any one of them is on the board, the other two must not be.
a. In how many ways can the Board of Directors be chosen?
b. After its selection, the Board of Directors will elect from its membership a president and a vice-president, how many possible outcomes are there for such an election?
Answer by VFBundy(438) About Me  (Show Source):
You can put this solution on YOUR website!
Ways to choose 5-man board with no restrictions = 12C5 = 12%21%2F%285%21%2A7%21%29 = 792

Ways to choose 5-man board where two or more are mutual enemies:
3C3 * 9C2 = 3%21%2F%283%21%2A0%21%29 * 9%21%2F%282%21%2A7%21%29 = 1 * 36 = 36
3C2 * 9C3 = 3%21%2F%282%21%2A1%21%29 * 9%21%2F%283%21%2A6%21%29 = 3 * 84 = 252
Total ways to choose 5-man board where two or more are mutual enemies = 36 + 252 = 288

Ways to choose 5-man board where two or more mutual enemies are not on the board = 792 - 288 = 504

a. In how many ways can the Board of Directors be chosen?

504 (from above)

b. After its selection, the Board of Directors will elect from its membership a president and a vice-president, how many possible outcomes are there for such an election?

5 * 4 = 20