In how many different ways can you place \(12\) chips in the squares of a \(4\times 4\) board so that
there is at most one chip in each square, and
every row and every column contains exactly three chips?
A book club with 37 members is reading the following books: “Brave New World", “Dracula" and “Flatland". Each member chose one of the books, though some people chose more than one book. We know that:
23 people chose “Brave New World";
18 people chose “Dracula";
26 people chose “Flatland";
7 people chose all three books.
How many people chose at least two books?
Jan wants to paint a map with \(95\) countries on it, where only one colour can be used for each country. He has \(33\) different colours to paint with and he must use each of yellow, blue, green, purple and red at least once. How many ways of painting the map are there?
A \(5\times5\) grid is given with \(25\) coins, each red on one side and white on the other. coins are placed on the grid one at a time. When a counter is placed, any coins on neighbouring squares are flipped. Two squares are neighbours if they share a side (not just a corner). The aim is to finish with all coins showing red. During the process coins may flip several times. How many flips occur in total?
The entire plane is coloured using two colours: red and blue. Prove that there must exist two points of the same colour that are exactly \(1\) meter apart.