# Combinatorial Analysis

#### math951

Delegates from 10 countries, including Russia, France, England, and the United States, are to be seated in a row. How many different seating arrangements are possible if the French and English delegates are to be seated next to each other and the Russian and U.S. delegates are not to be next to each other?

2*9!*2*2*8!

That is the answer. How do they get 2*2*8?

#### Plato

MHF Helper
Delegates from 10 countries, including Russia, France, England, and the United States, are to be seated in a row. How many different seating arrangements are possible if the French and English delegates are to be seated next to each other and the Russian and U.S. delegates are not to be next to each other? 2*9!*2*2*8! That is the answer. How do they get 2*2*8?
The block $$\displaystyle \boxed{EF}$$ can be arranged two ways
That block and six others can be used to separate $$\displaystyle U$$ from $$\displaystyle R$$.
Seven separators create eight places to put the $$\displaystyle U~\&~R$$ .
Thus there are $$\displaystyle 8\cdot 7$$ places for the $$\displaystyle U~\&~R$$.
There are the $$\displaystyle 2$$ ways to arrange the block and
$$\displaystyle 7!$$ ways to arrange the separators.
I get a total of $$\displaystyle 2\cdot 7!\cdot 8\cdot 7$$
I disagree with your given answer.
Lets review: Two ways to form the block; seven factorial ways to arrange the separators; eight places for the $$\displaystyle R$$; then seven places to put the $$\displaystyle U$$.

#### joshuaa

i would say 4*7!

Russia, France, England, and United Sates have always to be together in a row. This means we cannot put any country between any two of them. Moreover, we cannot put Russia and US next to each other.

So, we have four arrangements for the 4 countries

RFEU, REFU, UFER, UEFR

Let us call the four countries a Big country. So now we have a Big country and 6 other countries. They are 7 countries and their arrangements should be 7!

so the answer should be 4*7!

#### Plato

MHF Helper
i would say 4*7!
Russia, France, England, and United Sates have always to be together in a row. This means we cannot put any country between any two of them. Moreover, we cannot put Russia and US next to each other.
So, we have four arrangements for the 4 countries
RFEU, REFU, UFER, UEFR
Let us call the four countries a Big country. So now we have a Big country and 6 other countries. They are 7 countries and their arrangements should be 7! so the answer should be 4*7!
To Joshua: Did you actually read the post? We seat ten people in a row, with only two conditions:
1) two of the ten must be seated together side-by-side and
2) a different two of the ten must not be seated together(i.e. separated).
How else can it be read?

#### joshuaa

My bad, read it as the 4 countries were always together.

Plato answer is Correct

chip
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