Question

Одиннадцать футболистов строятся перед началом матча. Первым – обязательно капитан, вторым – обязательно вратарь, а остальные – случайным образом. Сколько существует способов построения?

Answer 1

Найти нужно перестановки 9 футболистов, т.к. капитан и вратарь уже стоят на своих местах.
Р₉=9!=1×2×3×4×5×6×7×8×9=362880 способов построения.
Ответ: 362880.

Answer 2

Calculation of the Number of Ways to Arrange the Football Players

To calculate the number of ways to arrange the eleven football players before the start of the match, we need to consider the positions of the captain, the goalkeeper, and the remaining players.

1. The captain is always positioned first. 2. The goalkeeper is always positioned second. 3. The remaining players can be arranged in the remaining nine positions in any order.

To calculate the number of ways to arrange the remaining players, we can use the concept of permutations. The number of permutations of n objects taken r at a time is given by the formula:

P(n, r) = n! / (n - r)!

In this case, we have nine remaining players to arrange in nine positions, so the number of ways to arrange them is:

P(9, 9) = 9! / (9 - 9)! = 9! / 0! = 9! = 362,880

Therefore, there are 362,880 ways to arrange the eleven football players before the start of the match.