Ten players were entered into a badminton tournament. The first round consisted of 5 matches, with each player in one match. In how many different ways could the 10 players be matched against each other?
There are again some adventurers standing in a queue to see a dragon’s treasure. This time, there are more of them – \(26\). The rules have changed slightly, they still enter exactly in the order they are queuing, but they now have to divide themselves into \(5\) groups, and some of the groups can be empty, do not consist of any adventurers at all. In how many ways can they do that now?
Problems often involve a protagonist, a quest and a story. In combinatorics, stories can help us prove identities and formulas, that would be difficult to prove otherwise. Here, you can write your own story, which will show that the following statement is always true:
The number of ways we can choose \(k\) out of \(n\) items is equal to the number of ways we can choose \(k\) out of \(n-1\) objects PLUS the number of ways in which we can choose \(k-1\) out of \(n-1\) objects.
Can you cover a \(10 \times 10\) board using only \(T\)-shaped tetraminoes?
Can you cover a \(10 \times 10\) square with \(1 \times 4\) rectangles?
Two opposite corners were removed from an \(8 \times 8\) chessboard. Can you cover this chessboard with \(1 \times 2\) rectangular blocks?
One small square of a \(10 \times 10\) square was removed. Can you cover the rest of it with 3-square \(L\)-shaped blocks?
A \(7 \times 7\) square was tiled using \(1 \times 3\) rectangular blocks. One of the squares has not been covered. Which one can it be?
Can you cover a \(13 \times 13\) square using two types of blocks: \(2 \times 2\) squares and \(3 \times 3\) squares?
Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.