Question

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?

Answer 1

\(\large \color{black}{\begin{align}
& \normalsize \text{ There are 20 people among whom two are sisters. }\hspace{.33em}\\~\\
& \normalsize \text{ Find the number of ways in which we can arrange }\hspace{.33em}\\~\\
& \normalsize \text{ them around a circle so that there is exactly one}\hspace{.33em}\\~\\
& \normalsize \text{ person between the two sisters?}\hspace{.33em}\\~\\
\end{align}}\)

Answer 2

Think of 20 slots laid out like this



I can't fit all 20, but you get the idea

Answer 3

the first and third slots are locked up by the 2 sisters

Answer 4

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

Answer 5

is answer 2*18!

Answer 6

yeah

Answer 7

ok thanks