1. **Problem Statement:** We need to find the number of different combinations of candies taken from a bowl containing 7 identical kiwi, 10 identical cherry, and 7 identical orange candies, with the condition that at least 3 kiwi candies must be included. 2. **Understanding the problem:** Since the candies of each flavor are identical, this is a problem of counting combinations with restrictions. 3. **Variables:** Let $k$ be the number of kiwi candies chosen, $c$ the number of cherry candies, and $o$ the number of orange candies. 4. **Constraints:** - $3 \leq k \leq 7$ (minimum 3 kiwi candies, maximum 7 available) - $0 \leq c \leq 10$ - $0 \leq o \leq 7$ 5. **Counting combinations:** For each flavor, the number of ways to choose candies is the number of possible quantities: - Kiwi: $7 - 3 + 1 = 5$ ways (choosing 3,4,5,6, or 7) - Cherry: $10 - 0 + 1 = 11$ ways (choosing 0 to 10) - Orange: $7 - 0 + 1 = 8$ ways (choosing 0 to 7) 6. **Total combinations:** Since choices for each flavor are independent, multiply the number of ways: $$5 \times 11 \times 8 = 440$$ 7. **Final answer:** There are **440** different combinations of candies that can be taken with at least 3 kiwi candies included.