The Prime Minister of XYZ calls a meeting of Chief Ministers of the states whose state boundaries touch ZZZ (there are total 4 such states) to discuss on the terrorism. In how many ways can they seat themselves at a round table if two of the four states Chief Ministers choose to sit together?
closed as off-topic by Pragabhava, JMoravitz, projectilemotion, Namaste, JonMark Perry Mar 11 '17 at 2:31
This question appears to be off-topic. The users who voted to close gave this specific reason:
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To summarize what I said in the comments above:
Approach by multiplication principle. Begin by arbitrarily assigning labels to the people to be seated calling them $A,B,C,D,E$ where $A$ and $B$ are the ones who wish to sit together.
Step 0: Seat $A$ at the table in an arbitrarily selected position. Important: This step does not contribute to our multiplication principle because making different choices here does not actually lead to different outcomes
Step 1: Since $B$ wants to sit next to $A$, choose whether $B$ is sitting to the right of $A$ or to the left. ($2$ options)
Step 2: Moving clockwise around the table from where $A$ and $B$ are currently sitting in some order, fill the next available empty seat with one of the three remaining people. Choose which person fills that seat. ($3$ options)
Step 3: Continuing clockwise, pick who occupies the next empty seat. ($2$ options)
Step 4: Continuing clockwise, pick who occupies the final empty seat ($1$ option)
Apply multiplication principle and reach a final conclusion.
Let's say we have 5 people A,B,C,D,E that sit at this table, and that A and B want to sit next to each other.
I'll assume that 1) everyone shifting one seat clockwise does not count as a different arrangement (though everyone sits in a different chair when we do so, everyone will still have the same person to their left and the same person to their right).
I'll also assume that 2) a mirror arrangement does count as a different arrangement (though everyone will still have the same two people they sit next to, the person that sat on their left will not on their right, and vice versa).
With this, let's start with seating A and B. Because of 1), it doesn't matter which seat I put A, but because of 2), B can be on the left of A or on the right, so there are 2 possibilities for that.
Once A and B are seated, let's put a third person clockwise from them, for which there are 3 possibilities, and for the last two people we then have 2 options left.
So: we get 2 (arranging A and B) * 3 (seat 3rd person) * 2 (seat last two people) = 12 possible arrangements