Question

Six teachers and six students have to sit around a circular table such that there is a teacher between any two students. The number of ways in which they can sit is:
A. $6! \times 6!$
B. $5! \times 6!$
C. $5! \times 5!$
D. None of these

Answer 1

Hint: We are required to find the number of ways of seating six teachers and six students in such a way that there is a teacher in between any two students. This means that no students can sit together. So, we will apply the concepts of permutations and combinations and use the factorial formula to find the number of arrangements.

Complete step by step solution:
There is a teacher between any two students means that no two students are together.
So, we first arrange the six students around the circular table and then arrange the six teachers such that there is a teacher between any two students.
So, we know that n things can be arranged in a circular arrangement in $\left( {n - 1} \right)!$ ways.
So, six students can be arranged in a circular arrangement for $5!$.
Now, we have to arrange the six teachers at six vacant positions in the circle. So, the number of ways of arranging six teachers at six positions is $6!$.
According to the multiplication rule of counting, if we have m ways to do a certain thing and n ways to do another thing, then there are a total of $m \times n$ ways to do both the things together.
Hence, the number of arrangements $ = 5! \times 6!$.

Option ‘B’ is correct

Note: One should know about the principle rule of counting or the multiplication rule. If we have to arrange n things out of which r things, then the number of ways of doing so are $\left( {\dfrac{{n!}}{{r!}}} \right)$. This formula can be used while arranging identical objects.