## mathmath333 one year ago 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?

1. 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}}

2. jim_thompson5910

Think of 20 slots laid out like this |dw:1442808790743:dw| I can't fit all 20, but you get the idea

3. jim_thompson5910

the first and third slots are locked up by the 2 sisters |dw:1442808843579:dw|

4. 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

5. mathmath333

6. jim_thompson5910

yeah

7. mathmath333

ok thanks

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