There are just two ways in which $5$ may be expressed as the sum of two different positive (non-zero) integers, namely $5 = 4 + 1 = 3 + 2$. In how many ways, $9$ can be expressed as the sum of two different positive (non-zero) integers?

### Concept & Logic The problem asks for the number of ways to partition a number into two distinct positive integers. To find all unique pairs, we systematically list pairs starting from the highest possible first integer and lowest second integer, stopping when pairs repeat or become identical. ### Step-by-Step Solution **Given:** * Target sum = $9$ * Conditions: Two numbers must be positive (non-zero) and different. **Calculation / Deduction:** Let the two integers be $x$ and $y$ such that $x + y = 9$ and $x > y > 0$. Let's list the possible values: 1. If $y = 1$, then $x = 8$ ($8 + 1 = 9$) 2. If $y = 2$, then $x = 7$ ($7 + 2 = 9$) 3. If $y = 3$, then $x = 6$ ($6 + 3 = 9$) 4. If $y = 4$, then $x = 5$ ($5 + 4 = 9$) If we continue to $y = 5$, $x = 4$, this is the same pair as $(5, 4)$ just reversed. Since the order of addition does not create a new set of numbers, we stop here. Total valid pairs = $4$. ### Exam Strategy & Shortcut For a given integer $N$, the number of ways to express it as the sum of two distinct positive integers is given by $\frac{N-1}{2}$ if $N$ is odd, and $\frac{N-2}{2}$ if $N$ is even. Here $N = 9$ (odd), so the number of ways is $\frac{9-1}{2} = 4$. This formula gives the answer instantly without listing. ### Common Pitfall A common mistake is treating $(8+1)$ and $(1+8)$ as two different ways. The question asks for the combination of two numbers, not permutations, so order doesn't matter. Another error is including zero ($9+0$), but the question specifically restricts to non-zero integers. ### Final Answer Therefore, the correct answer is 4.