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?
Find how many \(5\)-digit numbers contain only one \(0\) and one \(1\) in their decimal representation
Mary, who wrote an online computer game, plans to assign every user a unique password.
a) Currently, she wants to have passwords that are \(8\) symbols long and are all made from lowercase letters \(a,b,c\). How many different passwords can she generate?
b) Her friend suggested a modification: use 8-symbol words made from letters \(a,b,c\), with exactly one capital letter, and also include one digit (\(0-9\)). How many times more passwords can she generate now?
How many numbers in the set \(\{ 1,2,3, \dots, 1000 \}\) can be expressed as a sum of \(6\) consecutive integer numbers?