vipulgoyal wrote:in how many ways can 3 couples sit in a circular table such that no pair of husband wife are opposite to each other ?
In circular arrangements, the first person seated is irrelevant.
What we need to count is the number of ways to arrange the OTHER people RELATIVE to the first person seated.
Let A = the first person seated.
After A is seated:
Number of options for the seat opposite A = 4. (Of the 5 remaining people, anyone but A's spouse.)
Number of options for the seat to the left of A = 4. (Any of the 4 remaining people.)
Number of options for the opposite seat = 2. (Of the 3 remaining people, anyone but the spouse of person to A's left.)
Number of options for the seat to the right of A = 2. (Either of the 2 remaining people.)
Number of options for the last seat = 1. (Only 1 person left.)
To combine these options, we multiply:
4*4*2*2*1 =
64.
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