Let the 3 couples be A and B, C and D, E and F.smanstar 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 ?
1) 60
2) 100
3)64
4) 96
With circular arrangements, where the FIRST person sits is IRRELEVANT.
All that matters is the number of ways to arrange the remaining people RELATIVE to the first person.
Case 1: C does not sit opposite A or B
Once A is seated, the number of options for B = 4. (Of the 5 remaining seats, 4 are not opposite A.)
The number of options for C = 2. (Of the 4 remaining seats, 2 are not opposite A or B.)
Number of options for D = 2. (Of the 3 remaining seats, 2 are not opposite C.)
Number of options for E = 2. (Either of the 2 remaining seats.)
Number of options for F = 1. (Only 1 seat left.)
To combine these options, we multply:
4*2*2*2*1 = 32.
Case 2: C sits opposite A or B
Once A is seated, the number of options for B = 4. (Of the 5 remaining seats, 4 are not opposite A.)
Number of options for C = 2. (Must be opposite A or B.)
Of the 3 remaining seats, one is opposite A or B.
If D occupies this seat, then C and D will be opposite A and B, forcing E and F into opposite seats.
Thus, D must occupy one of the other 2 seats.
Number of options for D = 2.
Number of options for E = 2. (Either of the 2 remaining seats.)
Number of options for F = 1. (Only 1 seat left.)
To combine these options, we multply:
4*2*2*2*1 = 32.
Total options = 32+32 = 64.
