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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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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