1. Problem 1: Arranging Books on a Shelf Suppose you have 5 different books and you want to arrange 3 of them on a shelf. How many different ways can you arrange these 3 books? Formula: The number of permutations of choosing $r$ objects from $n$ distinct objects is given by $$P(n,r) = \frac{n!}{(n-r)!}$$ Explanation: Here, order matters because arranging books in different orders counts as different permutations. Calculation: $$P(5,3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60$$ So, there are 60 different ways to arrange 3 books out of 5. 2. Problem 2: Assigning Roles in a Team Imagine you have 4 candidates and 2 different positions to fill: President and Secretary. How many ways can you assign these positions? Formula: Since the positions are distinct, order matters, so use permutations: $$P(4,2) = \frac{4!}{(4-2)!}$$ Calculation: $$P(4,2) = \frac{4!}{2!} = \frac{24}{2} = 12$$ There are 12 ways to assign the President and Secretary roles from 4 candidates.