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An appointments board must consist of the Head of School and exactly 4 other members of the School chosen from a group consisting of 3 men and 4 women. If the overall board of 5 people must include at least 2 women and at least 2 men and the Head of School is male, how many different makeups are possible?

I have attempted this question and I got 2 different makeups is possible- two men (including Head of School) & three women and three men (including Head of School) & two women? Could someone tell me if I'm correct or if I'm taking too simple an approach to this question?

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    $\begingroup$ People are distinguishable. $\endgroup$ – Graham Kemp Dec 11 '17 at 11:02
  • $\begingroup$ But they have the same positions available to them so does it matter that they're distinguishable? $\endgroup$ – Derek Dec 11 '17 at 11:04
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    $\begingroup$ The groups that include Bob have different make-ups from those that do not. $\endgroup$ – Graham Kemp Dec 11 '17 at 11:07
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I have attempted this question and I got 2 different makeups is possible- two men (including Head of School) & three women and three men (including Head of School) & two women? Could someone tell me if I'm correct or if I'm taking too simple an approach to this question?

You are correct so far.   However, remember that people are distinguishable objects.   Now count how many distinct ways there are to select those combinations from the head of school, 3 other men, and 4 women.

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