Question 1145543
Ways to choose 5-man board with no restrictions = 12C5 = {{{12!/(5!*7!)}}} = 792
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Ways to choose 5-man board where two or more are mutual enemies:
3C3 * 9C2 = {{{3!/(3!*0!)}}} * {{{9!/(2!*7!)}}} = 1 * 36 = 36
3C2 * 9C3 = {{{3!/(2!*1!)}}} * {{{9!/(3!*6!)}}} = 3 * 84 = 252
Total ways to choose 5-man board where two or more are mutual enemies = 36 + 252 = 288
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Ways to choose 5-man board where two or more mutual enemies are not on the board = 792 - 288 = 504
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a. In how many ways can the Board of Directors be chosen? 
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504 (from above)
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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?
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5 * 4 = 20