How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,
a) if each number can occur only once?
b) if each number can occur several times?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Find the number of rectangles made up of the cells of a board with \(m\) horizontals and \(n\) verticals that contain a cell with the coordinates \((p, q)\).
Prove that there is no graph with five vertices whose degrees are equal to 4, 4, 4, 4, 2.
Prove that there exists a graph with 2n vertices whose degrees are \(1, 1, 2, 2, \dots , n, n\).
a) they have 10 vertices, the degree of each of which is equal to 9?
b) they have 8 vertices, the degree of each of which is equal to 3?
c) are they connected, without cycles and contain 6 edges?
Prove that a graph, in which every two vertices are connected by exactly one simple path, is a tree.
Prove that, in a tree, every two vertices are connected by exactly one simple path.
Prove that there is a vertex in the tree from which exactly one edge emerges (such a vertex is called a hanging top).
In a graph, all the vertices have degree of 3. Prove that there is a cycle in it.