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I came across $2$ questions:

  1. A jury pool of $20$ people are called to a courthouse. How many ways are there to select $12$ to serve as jury?

  2. A jury pool of $20$ people are called to a courthouse. How many ways are there to select $11$ to serve as jury, and $1$ to serve as jury foreman?

Can someone give me an intuitive explanation on how to think about this? Mathematically I know the calculation returns different values but I don't really understand why. It can't be because of the distinction between "jury" and "jury foreman", right? Ultimately we're still selecting $12$ people from a pool of $20$ isn't it?

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    $\begingroup$ It would help people give you good answers if you could explain why you think they should be equal $\endgroup$ – postmortes Jul 25 at 8:32
  • $\begingroup$ @postmortes I have edited my question, hope it's much clearer! $\endgroup$ – Hannah Tang Jul 25 at 8:37
  • $\begingroup$ Given your revised question, you should change your title. Do you wish to ask why $\binom{20}{12} \neq \binom{20}{11}\binom{9}{1}$ or why the number of ways to pick a jury differs from the number of ways of choosing a jury with a foreman? $\endgroup$ – N. F. Taussig Jul 25 at 8:41
  • $\begingroup$ Do you mean $$\binom{6}{3}=\binom{6}{2}\binom{4}{1}$$? $\endgroup$ – Dr. Sonnhard Graubner Jul 25 at 8:41
  • $\begingroup$ @Dr.SonnhardGraubner That is what she means. However, she has revised her question since you first read it. $\endgroup$ – N. F. Taussig Jul 25 at 8:42
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The difference lies in making the decision on who is the jury foreman. In the first case, all the 12 selected people are equal. In the second, one of the selected 12 needs to be designated the jury foreman, which can be then done in 12 different ways. So, the answer to the second should be 12 times the answer to the first.

One can also verify that: $$12\times\binom{20}{12}=\binom{20}{11}\binom{9}{1}$$

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    $\begingroup$ Your equation should be $12\times \binom{20}{12}=\binom{20}{11}\binom{9}{1}$ rather than $\binom{20}{12}=12\times\binom{20}{11}\binom{9}{1}$ as you want to designate one of the 12 jury members as foreman. $\endgroup$ – parafoo Jul 25 at 9:13

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