Прошу помочь с задачей A cube made of 29,791 small cubes gets all of its sides painted. Let S be the set of all cubes enclosed in the 29,791 small cubes structure that are made up of at least one small cube. A random element in S will be drawn. Find the expected value of number of completely painted sides of this randomly selected cube.
Куб, состоящий из 29 791 маленьких кубиков, окрашивает все свои стороны. Пусть S - набор всех кубиков, заключенных в структуру 29 791 маленьких кубиков, которые состоят по крайней мере из одного маленького кубика. Будет нарисован случайный элемент в S. Найдите ожидаемое значение количества полностью закрашенных сторон этого случайно выбранного куба.

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To find the expected value of the number of completely painted sides of the randomly selected cube, we need to calculate the probability of each possible number of completely painted sides and then multiply each probability by the corresponding number of completely painted sides and sum them up.

The total number of small cubes in the cube is 29,791. Since a cube has 6 sides, there are 6 faces for each small cube. Therefore, there are a total of 29,791 * 6 = 178,746 faces in the entire structure.

For a cube made up of n small cubes, the probability that a randomly selected cube has k completely painted sides is given by the formula: (6 choose k) * (n-6 choose n-k) / (n choose 6), where (a choose b) represents the number of ways to choose b items from a.

Now, we need to calculate the expected value:

E(x) = Σ( k * P(X=k) ) for all k = 0, 1, 2, 3, 4, 5, 6

E(x) = Σ( k (6 choose k) (29791-6 choose 29791-k) / (29791 choose 6) ) for all k = 0, 1, 2, 3, 4, 5, 6

Calculating each term and summing them up will give us the expected value of the number of completely painted sides of the randomly selected cube.