1. **Problem statement:** (a)(i) A restaurant offers a three-course meal with 4 starters, 6 main courses, and 8 desserts. We need to find the total number of different three-course meal combinations. (a)(ii) Jack finds the restaurant has 4 starters, 6 main courses, but fewer desserts, resulting in 120 different meal combinations. We need to find how many desserts are still available. 2. **Formula used:** The total number of combinations when choosing one item from each category is given by the product rule: $$\text{Total combinations} = (\text{number of starters}) \times (\text{number of mains}) \times (\text{number of desserts})$$ 3. **Solution for (i):** $$\text{Total combinations} = 4 \times 6 \times 8 = 192$$ So, there are 192 different three-course meal combinations available. 4. **Solution for (ii):** Let the number of desserts available to Jack be $d$. Given: $$4 \times 6 \times d = 120$$ Simplify: $$24 \times d = 120$$ Divide both sides by 24: $$\cancel{24} \times d = \frac{120}{\cancel{24}}$$ $$d = 5$$ So, Jack has 5 desserts available. **Final answers:** (i) 192 combinations (ii) 5 desserts