1. **Problem:** Identify whether the following are permutations or combinations and determine the number of possible ways. **a)** A popular brand of pen is available in three colors (red, green, blue) and four tips (bold, medium, fine, micro). How many different choices of pens do you have with this brand? 2. **Formula and Explanation:** - Since each pen choice is a combination of one color and one tip, and order does not matter here, this is a multiplication of independent choices. - Number of ways = number of colors $ \times $ number of tips. 3. **Calculation:** - Number of colors = 3 - Number of tips = 4 - Total choices = $3 \times 4 = 12$ 4. **Answer:** There are 12 different choices of pens. 1. **Problem:** A corporation has ten members on its board of directors. In how many ways can it elect a president, vice-president, secretary, and treasurer? 2. **Formula and Explanation:** - This is a permutation problem because the positions are distinct and order matters. - Number of ways to choose 4 officers from 10 members is given by permutation formula: $$P(n, r) = \frac{n!}{(n-r)!}$$ where $n=10$ and $r=4$. 3. **Calculation:** $$P(10,4) = \frac{10!}{(10-4)!} = \frac{10!}{6!} = 10 \times 9 \times 8 \times 7 = 5040$$ 4. **Answer:** There are 5040 ways to elect the four officers. **Final answers:** - a) 12 choices (Combination of independent features) - b) 5040 ways (Permutation of distinct positions)