Question

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

Answer 1

Possible number of arrangements = 564480

Explanation:

Total no. of delegates = 10

Delegates to sit together = 2 (French and English) .

We can consider French and English delegate as a single entity in this give situation

In this situation total entities left = 9

Possible number of arrangements = 9!

Possible arrangements for French and English delegates = 2!=2 --(1)

= (FE, EF).

At this moment total possible arrangements for all delegates = 9!×2! ----- (2)

It is also given Russian and American delegates are not to be seated next to each other then to calculate possible arrangements with this condition, we calculate total ways in which US and Russian delegates sit next to each other and subtract it from (2)

With condition of French and English delegates total entities left =9

Now if Russian and U.S. delegates are to be seated next to each other than entities left = 8 ---(3)

Following the strategy of French and English, possible arrangements for Russian and U.S. delegates = 2! = 2 (RU, UR) ---(4)

Considering conditions (1), (3) and (4)

Possible arrangements = 8!2!2! ---- (5)

subtracting (5) from (2) we can get possible number of seating arrangements when French and English delegates sit together and Russian and U.S do not sit together, i.e.

Possible number of arrangements = 9!2! - 8!2!2! = 564480