excecutives from 25 student clubs, one male and one female from each are attending a workshop on student violence. how many ways can a commitee be set up of 5 men and 7 women if only one male or female from each club can be selected

  • $\begingroup$ So if a man is chosen from a club, the woman from that same club cannot be on the committee? $\endgroup$ – turkeyhundt Jul 13 '16 at 5:28
  • $\begingroup$ That is how I interpreted it. $\endgroup$ – K. Jiang Jul 13 '16 at 5:30
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    $\begingroup$ As an alternative to the answer by K. Jiang, the men can be chosen in $\binom{25}{5}$ ways and for each choice the women can be chosen in $\binom{20}{7}$ ways. Multiply. The answer looks different, but is the same. $\endgroup$ – André Nicolas Jul 13 '16 at 6:02

Use constructive counting. First select which $12$ of the $25$ clubs will have representatives. This can obviously be done in $\dbinom{25}{12}$ ways. Now that we have our subset, we must choose $5$ of these $12$ clubs to be represented by the male. This can be done in $\dbinom{12}{5}$ ways. Our final answer is $$\dbinom{25}{12} \times \dbinom{12}{5}$$ $$= \boxed{4,118,637,600}$$ possible committees.

  • $\begingroup$ okay, thank you so much! so the women would not be counted at all? $\endgroup$ – krystal Jul 13 '16 at 5:36
  • $\begingroup$ You are welcome! The women come from the remaining clubs from the $12$ selected clubs. You can see that we can actually count the ways to chose the women from the $12$ clubs, which is $\dbinom{12}{7},$ which is the same as choosing the men. But it would be incorrect to multiply because once we have selected one, the others are also automatically selected. $\endgroup$ – K. Jiang Jul 13 '16 at 5:49
  • $\begingroup$ oh okay, that makes sense! thanks again! $\endgroup$ – krystal Jul 13 '16 at 5:51
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    $\begingroup$ @krystal We could also write K. Jiang's answer as $$\binom{25}{12} \binom{12}{5}\binom{7}{7}$$ where $\binom{25}{12}$ represents the number of ways of choosing the clubs from which the representatives are drawn, $\binom{12}{5}$ represents the number of ways of choosing a club from which to draw a male representative from the $12$ selected clubs, and $\binom{7}{7}$ represents the number of ways of choosing the clubs from which to draw female representative from the remaining seven selected clubs from which a male representative has not been drawn. $\endgroup$ – N. F. Taussig Jul 13 '16 at 11:56

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