Problems

Paloma wrote digits from \(0\) to \(9\) in each of the \(9\) dots below, using each digit at most once. Since there are \(9\) dots and \(10\) digits, she must have missed one digit.

In the triangles, Paloma started writing either the three digits at the corners added together (the sum), or the three digits at the corners multiplied together (the product). She gave up before finishing the final two triangles.

What numbers could Paloma have written in the interior of the red triangle? Demonstrate that you’ve found all of the possibilities.

How many subsets are there of \(\{1,2,...,10\}\) (the integers from \(1\) to \(10\) inclusive) containing no consecutive digits? That is, we do count \(\{1,3,6,8\}\) but do not count \(\{1,3,6,7\}\).
For example, when \(n=3\), we have \(8\) subsets overall but only \(5\) contain no consecutive integers. The \(8\) subsets are \(\varnothing\) (the empty set), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,3\}\), \(\{1,2\}\), \(\{2,3\}\) and \(\{1,2,3\}\), but we exclude the final three of these.

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