Question

The chief minister of 11 states of India met to discuss the language problem. The number of ways they can seat themselves at a round table so that the Punjab and Madras chief ministers sit together is,
A.$725760$
B.$725748$
C.$725778$
D.None of these

Answer 1

First of all we have to arrange the chief minister of 11 states in a round table such that chief ministers Punjab and Madras sit together. For that we count chief ministers Punjab and Madras as one entity so we have to arrange 10 members at a round table. For that we are using the fact that n people can be sited in a round table in $\left( {n - 1} \right)!$ways. More over the Punjab and Madras chief ministers can itself be sited in 2 ways. So multiply the number of ways 10 members can sit at the round table with $2!$in order to get the answer.

Complete step-by-step answer:
To find the number of ways we will use permutation.
As chief ministers of Punjab and Madras will always sit together, we will count them as one entity.
So, in total we have 10 entity that is 9 chief ministers $+$1(Punjab C.M.+ Madras C.M)
As we know n people can be sited in a round table in $\left( {n - 1} \right)!$ways.
So, 10 entities can be sited in a round table in $(10 - 1)!$ways and the Punjab and Madras chief ministers can itself be sited in 2 ways.
So, we have
$(10 - 1)! \times 2!$
$\Rightarrow 9! \times 2!$
$\Rightarrow 9 \times 8 \times 7 \times 6 \times 4 \times 3 \times 2 \times 1 \times 2 = 725760$
Thus, we can say that the number of ways they can seat themselves at a round table so that the Punjab and Madras chief ministers sit together is$725760$.
Hence, option A. $725760$is the correct answer.

Note: The arrangement of objects in a circular manner are called circular permutation or one can say that circular permutation is the total number of ways in which n distinct objects can be arranged around a fixed circle.