Given N(A)=8, N(B)=12, N(F)=12, N(A intersect F)=5, N(B intersect F)=6, N(only A)=2, N(A intersect B intersect F)=3, and total=20, find N(only B).

The number of people who liked all three foods is given:

• People who liked Ampese and Fufu only: $$N(\text{only } A \cap F) = N(A \cap F) - N(A \cap B \cap F) = 5 - 3 = 2$$
• People who liked Banku and Fufu only: $$N(\text{only } B \cap F) = N(B \cap F) - N(A \cap B \cap F) = 6 - 3 = 3$$
• To find people who liked Ampese and Banku only, we use $N(A)$: We know $N(A) = N(\text{only } A) + N(\text{only } A \cap B) + N(\text{only } A \cap F) + N(A \cap B \cap F)$. $$8 = 2 + N(\text{only } A \cap B) + 2 + 3$$ $$8 = 7 + N(\text{only } A \cap B)$$ $$N(\text{only } A \cap B) = 8 - 7 = 1$$

• People who liked Ampese only is given: $$N(\text{only } A) = 2$$
• People who liked Banku only: We know $N(B) = N(\text{only } B) + N(\text{only } A \cap B) + N(\text{only } B \cap F) + N(A \cap B \cap F)$. $$12 = N(\text{only } B) + 1 + 3 + 3$$ $$12 = N(\text{only } B) + 7$$ $$N(\text{only } B) = 12 - 7 = 5$$
• People who liked Fufu only: We know $N(F) = N(\text{only } F) + N(\text{only } A \cap F) + N(\text{only } B \cap F) + N(A \cap B \cap F)$. $$12 = N(\text{only } F) + 2 + 3 + 3$$ $$12 = N(\text{only } F) + 8$$ $$N(\text{only } F) = 12 - 8 = 4$$

Sum of all regions: $2 (\text{only A}) + 5 (\text{only B}) + 4 (\text{only F}) + 1 (\text{only A and B}) + 2 (\text{only A and F}) + 3 (\text{only B and F}) + 3 (\text{A and B and F}) = 20$.
This matches the total number of people in the house, so no one liked none of the foods.

a) Illustrate the information on a Venn diagram.
A Venn diagram with three overlapping circles (Ampese, Banku, Fufu) would show the following values in each region:

• Only Ampese: 2
• Only Banku: 5
• Only Fufu: 4
• Only Ampese and Banku: 1
• Only Ampese and Fufu: 2
• Only Banku and Fufu: 3
• Ampese, Banku, and Fufu: 3

b) Find the number of people who liked:

i) only fufu
The number of people who liked only fufu is $N(\text{only } F)$.

The number of people who liked only fufu is $\boxed{\text{4}}$.

ii) only one type of food
This is the sum of people who liked only Ampese, only Banku, and only Fufu.

The number of people who liked only one type of food is $\boxed{\text{11}}$.