## 37. International Conference

Four French, two representatives from the Republic of Ivory Coast, three English and four Swedes gathered at a conference to discuss forestry issues. The conference participants from the same country initially sat next to each other on a bench with 13 seats during the opening ceremony, and later, during the working sessions, at a round table with 13 seats.

In how many different ways can the conference participants sit next to each other in each of these two cases?
37. The number of ways in which 4 French people can sit next to each other on a bench is 4!; representatives from Ivory Coast can be arranged in 2! ways; the English - in 3! ways, and the Swedes - in 4! ways.

From this, we get that the number of ways in which all 13 people can sit on a bench, given that the order of the national delegations is fixed, is

$$
4! \cdot 2! \cdot 3! \cdot 4! = 6912
$$[^23]

The number of ways in which the 4 national delegations can be arranged among themselves on a bench is 4!

Therefore, the total number of possible arrangements of 13 people on a bench is

$$
4! \cdot 6912 = 165888
$$

The number of ways in which the national delegations can be arranged around a round table is, obviously, not 4!, but 3!. Therefore, the total number of possible arrangements of 13 people around a round table is

$$
3! \cdot 6912 = 41472
$$

Explanation. Let's denote the delegations participating in the conference with letters: F - French, C - Ivory Coast, E - English, S - Swedes. The possible arrangements of the delegations around a round table are: FCES, FCSE, FCEC, FSEC, FSCC, FSEC.

The total number of such arrangements is 6, or 3!. On a bench, we have 4 times more arrangements, since, for example, ECF and FSEC are two different orders of arrangement of the delegations, while around a round table, such two arrangements are no longer distinguishable from each other.