mathmath333
  • mathmath333
There are 20 people among whom two are sisters. Find the number of ways in which we can arrange them around a circle so that there is exactly one person between the two sisters?
Mathematics
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mathmath333
  • mathmath333
There are 20 people among whom two are sisters. Find the number of ways in which we can arrange them around a circle so that there is exactly one person between the two sisters?
Mathematics
schrodinger
  • schrodinger
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \normalsize \text{ There are 20 people among whom two are sisters. }\hspace{.33em}\\~\\ & \normalsize \text{ Find the number of ways in which we can arrange }\hspace{.33em}\\~\\ & \normalsize \text{ them around a circle so that there is exactly one}\hspace{.33em}\\~\\ & \normalsize \text{ person between the two sisters?}\hspace{.33em}\\~\\ \end{align}}\)
jim_thompson5910
  • jim_thompson5910
Think of 20 slots laid out like this |dw:1442808790743:dw| I can't fit all 20, but you get the idea
jim_thompson5910
  • jim_thompson5910
the first and third slots are locked up by the 2 sisters |dw:1442808843579:dw|

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jim_thompson5910
  • jim_thompson5910
we have 20-2 = 18 slots left with 18 people left to fill them so there are 18*17*16*...*3*2*1 = 18! ways to fill up the rest of the slots now because there are 2 ways for the sisters to sit in the first and third seats, we double that to say 2*18! total ways to arrange the people
mathmath333
  • mathmath333
is answer 2*18!
jim_thompson5910
  • jim_thompson5910
yeah
mathmath333
  • mathmath333
ok thanks

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