Mrs. F has invited several wives of delegates to the United Nations for an info

Mrs. F has invited several wives of delegates to the United
Nations for an informal luncheon. She plans to seat her 9
guests ina row such that each lady will be able to converse
with the person directly to her left and right. She has
prepared the following list.

Mrs. F speaks English only.

Mrs. G speaks English and French.

Mrs. H speaks English and Russian.

Mrs. J speaks Russian only.

Mrs. K speaks English only.

Mrs. L speaks French only.

Mrs. M speaks French and German.

Mrs. N speaks English and German.

Mrs. O speaks English only.

How many distinct seating arrangements are possible? Give
all possible seating arrangements.

Note that ABCD and DCBA are the same.

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Mrs. F has invited several wives of delegates to the United Nations for an informal luncheon. She p..

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126 distinct seating arrangements are possible.

Mrs. J and Mrs. H must be together and Mrs. J must be at the
end as Mrs. J speaks only Russian and Mrs. H is the only
other Russian speaker.

Mrs. L speaks only French and there are two others - Mrs. G
and Mrs. M - who speak French. Here there are 2 cases.

? CASE A : Mrs. L is at the other end

If Mrs. L is at the other end, either Mrs. G or Mrs. M must
seat next to her.

o CASE AA : Mrs. G seats next to Mrs. L

Then, Mrs. M must seat next to Mrs. G and Mrs. N must seat
next to Mrs. M. This is because Mrs. M speaks French and
German, and Mrs. N is the only other German speaker. Thus,
the possible seating arrangement is JHxxxNMGL, where x is
the English speakers. Mrs. F, Mrs. K and Mrs. O can be
arranged in remaining 3 positions in 3! different ways i.e.
6 ways.

o CASE AB : Mrs. M seats next to Mrs. L

If so, then either Mrs. N or Mrs. G must seat next to Mrs. M

? CASE ABA : Mrs. N seats next to Mrs. M

Thus, the possible seating arrangement is JHxxxxNML, where x
is the English speakers. Mrs. F, Mrs. G, Mrs. K and Mrs. O
can be arranged in remaining 4 positions in 4! different
ways i.e. 24 ways.

? CASE ABB : Mrs. G seats next to Mrs. M

Thus, the possible seating arrangement is JHxxxxGML, where x
is the English speakers. Mrs. F, Mrs. K, Mrs. N and Mrs. O
can be arranged in remaining 4 positions in 4! different
ways i.e. 24 ways.

? CASE B : Mrs. L does not seat at the end

It means that Mrs. G, Mrs. L and Mrs. M must seat together.
Also, Mrs. L must seat between Mrs. G and Mrs. M.

o CASE BA : Mrs. G seats left and Mrs. M seats right to Mrs.
L i.e. GLM

? CASE BAA : GLM is at the other end

Thus, the possible seating arrangement is JHxxxxGLM, where x
is the English speakers. Mrs. F, Mrs. K, Mrs. N and Mrs. O
can be arranged in remaining 4 positions in 4! different
ways i.e. 24 ways.

? CASE BAB : GLM is not at the other end

Then Mrs. N must seat next to Mrs. M. Now, we have a group
of four GLMN where Mrs. G and Mrs. N speak English. Thus,
the possible seating arrangement is JHxxxX, where x is the
individual English speakers and X is the group of four
females with English speakers at the both ends. Thus, there
are 4! different ways i.e. 24 ways.

o CASE BB : Mrs. M seats left and Mrs. G seats right to Mrs.
L i.e. MLG

Then, Mrs. N must seat next to Mrs. M. Now, we have a group
of four NMLG where Mrs. G and Mrs. N speak English. Thus,
the possible seating arrangement is JHxxxX, where x is the
individual English speakers and X is the group of four
females with English speakers at the both ends. Thus, there
are 4! different ways i.e. 24 ways.

Thus, total different possible seating arrangements are :

= 6 (case AA) + 24 (case ABA) + 24 (case ABB) + 24 (case
BAA) + 24 (case BAB) + 24 (case BB)

= 126 seating arrangements

Thus, 126 distinct seating arrangements are poosible.

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