Four married couples are at a banquet and are to be seated at a circular table. How many ways to arrange them if spouses must sit next to one another and Mr. Wang cannot sit next to Mrs. Teoh and Mrs. Rama.

Treating each couple as a single entity, there are 24 ways of arranging 4 different couples, forgetting for a moment whether a man sits on his wife's left or right. Here are all arrangements, where A=another couple (unnamed), R=Ramas, T=Teohs and W=Wangs:

ARTW ARWT ATRW ATWR AWRT AWTR

RATW RAWT RTAW RTWA RWAT RWTA

TARW TAWR TRAW TRWA TWAR TWRA

WART WATR WRAT WRTA WTAR WTRA

When the Wangs are positioned between the Teohs and the Ramas Mr Wang cannot sit next to Mrs Teoh or Mrs Rama (that's how I interpreted the question, since Mr Wang has to sit with his wife on one side or the other).

If M and F mean male and female then, for example, TM means Mr Teoh and TF means Mrs Teoh.

There are two seating arrangements for each couple MF or FM. If there were no constraints there would be 16 ways of arranging one of the 24 permutations of couples:

MFMFMFMF MFMFMFFM MFMFFMMF MFMFFMFM

MFFMMFMF MFFMMFFM MFFMFMMF MFFMFMFM

FMMFMFMF FMMFMFFM FMMFFMMF FMMFFMFM

FMFMMFMF FMFMMFFM FMFMFMMF FMFMFMFM

The constraints reduce this number. When the unnamed couple sits between the Teohs and the Ramas the Wangs are sandwiched between the Teohs and Ramas. Otherwise, the Wangs will only have either the Teohs or the Ramas next to them.

When the Wangs sit next to the unnamed couple then the Ramas or Teohs will be on the other side, and there are 4 positions that Mr Wang cannot be seated. That's 4 out the possible 16 male-female arrangements, leaving 12 that are permitted. Inspecting the 24 couple arrangements we see that of the 24, 16 have the 12 permitted arrangements: 16*12=192.

When the Wangs are sandwiched between the Teohs and Ramas only 8 out of 16 MF arrangements are permitted, and there are 8 out of 24 couple arrangements where the sandwich occurs. That's 8*8=64. Add this to the 192 we calculated before and we get 192+64=256. So that's how many seating arrangements there are satisfying the given constraints.

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