8. Using induction prove that the total number of ways of picking a pair of objects from a given set of \( n \) objects is \( \frac{n(n-1)}{2} \).

To prove that the total number of ways of picking a pair of objects from a given set of \( n \) objects is \( \frac{n(n-1)}{2} \) using mathematical induction, we will follow these steps: ### Step 1: Base Case We start by checking the base case for \( n = 2 \). When \( n = 2 \), the number of ways to choose 2 objects from 2 is: \[ \binom{2}{2} = 1 \] Now, we calculate \( \frac{n(n-1)}{2} \): \[ \frac{2(2-1)}{2} = \frac{2 \cdot 1}{2} = 1 \] Thus, the base case holds true. ### Step 2: Inductive Step Assume that the statement is true for some integer \( k \), i.e., the number of ways to choose 2 objects from \( k \) objects is: \[ \frac{k(k-1)}{2} \] We need to show that it holds for \( k + 1 \). When we have \( k + 1 \) objects, we can choose 2 objects in two ways: 1. Choose 2 objects from the first \( k \) objects. 2. Choose 1 object from the first \( k \) objects and 1 object from the new \( (k + 1)^{th} \) object. The number of ways to choose 2 objects from the first \( k \) objects is: \[ \frac{k(k-1)}{2} \] The number of ways to choose 1 object from the first \( k \) objects is \( k \), and we can pair it with the new \( (k + 1)^{th} \) object. Thus, the number of ways to choose 1 object from the first \( k \) and 1 from the new object is: \[ k \] ### Total Ways for \( k + 1 \) Combining both cases, the total number of ways to choose 2 objects from \( k + 1 \) objects is: \[ \frac{k(k-1)}{2} + k \] Now, we need to simplify this expression: \[ \frac{k(k-1)}{2} + k = \frac{k(k-1)}{2} + \frac{2k}{2} = \frac{k(k-1) + 2k}{2} = \frac{k^2 - k + 2k}{2} = \frac{k^2 + k}{2} = \frac{k(k + 1)}{2} \] ### Conclusion Thus, we have shown that: \[ \text{Number of ways to choose 2 objects from } (k + 1) \text{ objects} = \frac{(k + 1)k}{2} \] This completes the inductive step. Since both the base case and the inductive step have been proven, by the principle of mathematical induction, the statement is true for all integers \( n \geq 2 \). Therefore, the total number of ways of picking a pair of objects from a given set of \( n \) objects is: \[ \frac{n(n-1)}{2} \]