Problem 1

You have a fixed set of three distinct integers from the range 1 to 9 inclusive. You may use each of the three integers at most once and the standard arithmetic operations \(+, -, \times, \div\) to obtain real numbers. What is the maximum number of different real numbers that can be obtained?

While you may use each integer at most once, you can use each of the four operators more than once, or not at all, in each expression. Note that you may only use subtraction as a only as binary operator, i.e. given the integers \(3\), \(5\) and \(6\), then \(5 \times -3\) is not allowed, but you could have \(5 \times (3 - 6)\). You may not concatenate the integers; in the previous example, you are not allowed \(35\) or any other such combinations.