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How many ways can a president and a vice-president be chosen if they cannot be both men or both women?

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closed as off-topic by SBareS, JMP, P Vanchinathan, zhoraster, user91500 Mar 16 '17 at 6:18

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    $\begingroup$ Choose one man and one woman, then select which of them will be president. $\endgroup$ – Henning Makholm Mar 15 '17 at 16:06
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    $\begingroup$ Hint: How many ways can you choose a male president and a female vice-president? $\endgroup$ – robjohn Mar 15 '17 at 16:06
  • $\begingroup$ There are 6 ways to pick a man and 5 ways to pick a woman. Then there is one more choice: whom of them gets which post. $\endgroup$ – mathreadler Mar 15 '17 at 16:07
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The obvious alternative is simply to work through the two cases and add:

$$\underset{\text{Pres}}{\binom 51}\underset{\text{VP}}{\binom 61} + \underset{\text{Pres}}{\binom 61}\underset{\text{VP}}{\binom 51} = 5\cdot 6 + 6\cdot5 = 30+30=60$$

Note $\binom xy$ ("$x$ choose $y$") are binomial coefficients.

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  • $\begingroup$ same thinking. +1 $\endgroup$ – Harsh Kumar Apr 19 '17 at 16:50
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Total number of ways $$=\dbinom61 \times \dbinom51 \times 2!$$ $$=60~\text{ways}$$

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