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?

Answer 1

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.

Complete step-by-step answer:
The number of ways in which 7 people can form a ring or a circular arrangement, we can use the formula as mentioned in the hint which is as follows
\[\begin{align}
  & =\left( 7-1 \right)! \\
 & =6! \\
 & =720 \\
\end{align}\]
As mentioned in the question, no two Americans should be placed together.
For achieving this thing, we will first place all the Americans around the round table by using the formula given in the hint as follows
\[\begin{align}
  & =\left( 7-1 \right)! \\
 & =6! \\
 & =720 \\
\end{align}\]
Now, we will place the Englishmen in between the Americans that is we will place the Englishmen in the 7 places that is between every two Americans, there should be an Englishman which is done as follows
\[=7!\]
Now, using the fundamental theorem, the ways in which no two Americans can sit together is as follows
\[\begin{align}
  & =6!\times 7! \\
 & =720\times 5040 \\
 & =368800 \\
\end{align}\]
Hence, the number of ways in which the event can be done is 368800.

Note: The students can make an error if they don’t know the fundamental theorem or the formula for arranging n different things in a circular arrangement which is as follows
\[=\left( n-1 \right)!\]