The Tour de Clochemerle is not yet as big as the rival Tour de France. This year there were five riders, Arouet, Barthes, Camus, Diderot and Eluard, who took part in five stages. The winner of each stage got \(5\) points, the runner up \(4\) points and so on down to the last rider who got \(1\) point. The total number of points acquired over the five states was the rider’s score. Each rider obtained a different score overall and the riders finished the whole tour in alphabetical order with Arouet gaining a magnificent 24 points. Camus showed consistency by gaining the same position in four of the five stages and Eluard’s rather dismal performance was relieved by a third place in the fourth stage and first place in the final stage.
Where did Barthes come in the final stage?
Prove that the product of five consecutive integers is divisible by \(120\).
Norman painted the plane using two colours: red and yellow. Both colours are used at least once. Show that no matter how Norman does this, there is a red point and a yellow point exactly \(1\)cm apart.
Dario is making a pizza. He has the option to choose from \(3\) different types of flatbread, \(4\) different types of cheese and \(2\) different sauces. How many different pizzas can he make?
Determine the number of \(4\)-digit numbers that are composed entirely of distinct even digits.
There are \(10\) boys that need to be arranged in a line. Two of these boys are brothers, who need to have an even number of other boys between them. How many possible arrangements are there?
Steve, a student, has discovered that he lost most of his socks, and as a result, none of them match anymore. He has \(4\) black right socks, \(6\) blue right socks, \(8\) black left socks, and \(5\) blue left socks. Additionally, he has \(2\) pairs of blue trousers and \(3\) pairs of black trousers. Steve wants to ensure that his clothing items match in colour, so they desire to have left socks, right socks, and trousers of the same colour. How many different ways can Steve dress up given these conditions?
Determine the number of \(5\)-digit numbers that have only one odd digit and all other digits are even and distinct.
A group of \(4\) adults and \(5\) children is on a mountain hiking trip. At one point, the path becomes really narrow, and the hikers have to move in a line. They agreed that the line has to both start and end with an adult, for safety reasons. In how many ways can they arrange themselves?
A restaurant offers \(5\) choices of starter, \(10\) choices of the main course, and \(4\) choices of dessert. A customer can choose to eat just one course, or two different courses, or all three courses. Assuming that all food choices are available, how many different possible meals does the restaurant offer?