QUESTION

# In how many ways can 7 people form a ring? In how many ways can 7 Englishmen and 7 Americans sit down at a round table, no two Americans being together?

Hint: The formula for arranging n different things in a circular arrangement is given as follows $\left( n-1 \right)!$
Now, for no two Americans sitting together, we have to place an Englishman between every two Americans.

\begin{align} & =\left( 7-1 \right)! \\ & =6! \\ & =720 \\ \end{align}
\begin{align} & =\left( 7-1 \right)! \\ & =6! \\ & =720 \\ \end{align}
$=7!$
\begin{align} & =6!\times 7! \\ & =720\times 5040 \\ & =368800 \\ \end{align}
$=\left( n-1 \right)!$