In a house of 20 people, 8 liked Ampesi, 12 liked Banku, and 12 liked Fufu. 5 liked Ampesi and Fufu, 6 liked Banku and Fufu, 2 liked only Ampesi and Banku, and 3 liked all three. How many people liked

Let A be the set of people who liked Ampesi.
Let B be the set of people who liked Banku.
Let F be the set of people who liked Fufu.
The total number of people in the house is $|U| = 20$.

Given:
$|A| = 8$
$|B| = 12$
$|F| = 12$
$|A \cap F| = 5$
$|B \cap F| = 6$
$|(A \cap B) \setminus F| = 2$ (people who liked only Ampesi and Banku)
$|A \cap B \cap F| = 3$ (people who liked all three types of food)

Start with the intersection of all three sets:

• $|A \cap B \cap F| = 3$ (liked all three)

Next, calculate the regions for exactly two types of food:

• Liked only Ampesi and Banku (not Fufu): $|(A \cap B) \setminus F| = 2$ (given)
• Liked only Ampesi and Fufu (not Banku): $|(A \cap F) \setminus B| = |A \cap F| - |A \cap B \cap F| = 5 - 3 = 2$
• Liked only Banku and Fufu (not Ampesi): $|(B \cap F) \setminus A| = |B \cap F| - |A \cap B \cap F| = 6 - 3 = 3$

Now, calculate the regions for only one type of food:

• Liked only Ampesi: $|A \text{ only}| = |A| - (|(A \cap B) \setminus F| + |(A \cap F) \setminus B| + |A \cap B \cap F|)$ $|A \text{ only}| = 8 - (2 + 2 + 3) = 8 - 7 = 1$
• Liked only Banku: $|B \text{ only}| = |B| - (|(A \cap B) \setminus F| + |(B \cap F) \setminus A| + |A \cap B \cap F|)$ $|B \text{ only}| = 12 - (2 + 3 + 3) = 12 - 8 = 4$
• Liked only Fufu: $|F \text{ only}| = |F| - (|(A \cap F) \setminus B| + |(B \cap F) \setminus A| + |A \cap B \cap F|)$ $|F \text{ only}| = 12 - (2 + 3 + 3) = 12 - 8 = 4$

Finally, calculate the number of people who liked none of the foods:
Sum of all calculated regions: $1 + 4 + 4 + 2 + 2 + 3 + 3 = 19$
Number of people who liked none of the foods: $|U| - 19 = 20 - 19 = 1$

The values for each distinct region are:

• Only Ampesi: 1
• Only Banku: 4
• Only Fufu: 4
• Only Ampesi and Banku: 2
• Only Ampesi and Fufu: 2
• Only Banku and Fufu: 3
• All three (Ampesi, Banku, Fufu): 3
• None of the three foods: 1

From Step 2, the number of people who liked only Fufu is 4.

$$\text{Only Fufu} = \boxed{4}$$

This is the sum of people who liked only Ampesi and Banku, only Ampesi and Fufu, and only Banku and Fufu.

$$\text{Exactly two types} = |(A \cap B) \setminus F| + |(A \cap F) \setminus B| + |(B \cap F) \setminus A|$$ $$\text{Exactly two types} = 2 + 2 + 3 = \boxed{7}$$

This is the sum of people who liked only Ampesi, only Banku, and only Fufu.

$$\text{Only one type of food} = |A \text{ only}| + |B \text{ only}| + |F \text{ only}|$$ $$\text{Only one type of food} = 1 + 4 + 4 = \boxed{9}$$