1. **State the problem:** We have a group of 20 people. Among them, 13 like tea, 12 like coffee, and 3 like neither tea nor coffee. We want to find how many people like tea but not coffee. 2. **Identify known values:** - Total people, $N = 20$ - People who like tea, $|T| = 13$ - People who like coffee, $|C| = 12$ - People who like neither, $|N| = 3$ 3. **Calculate the number of people who like tea or coffee or both:** Since 3 people like neither, the number of people who like tea or coffee or both is: $$|T \cup C| = N - |N| = 20 - 3 = 17$$ 4. **Use the formula for union of two sets:** $$|T \cup C| = |T| + |C| - |T \cap C|$$ Substitute the known values: $$17 = 13 + 12 - |T \cap C|$$ 5. **Solve for the intersection:** $$|T \cap C| = 13 + 12 - 17 = 25 - 17 = 8$$ 6. **Find the number of people who like tea but not coffee:** This is the number of people in $T$ excluding those who also like coffee: $$|T \setminus C| = |T| - |T \cap C| = 13 - 8 = 5$$ **Final answer:** $$\boxed{5}$$ people like tea but not coffee.