In a bird sanctuary 41 different special of birds are being studied. Three large bird feeders are constructed, each providing a different type of bird feed. One feeder has sunflower seeds. A second feeder has a mixture of seeds, and the third feeder has small pieces of fruit. The following information was obtained.

20 species ate sunflower seeds
22 species ate the mixture
11 species ate the fruit
10 species ate the sunflower seeds and the mixture

4 species ate the sunflower seeds and the fruit
3 species ate the mixture and the fruit
1 species ate all three.

• How many species ate
– A. None of the foods
– B. The sunflower seeds, but neither of the other two foods?
– C. The mixture and the fruit, but not the sunflower seeds?
– D The mixture or the fruit, but not the sunflower seeds?
– E. Exactly one of the foods?

A. 23

B. 20
C. 3
D. 33
E. 53

To find the number of species that ate each combination of foods, we can use a Venn diagram. Let's label the three circles as S (sunflower seeds), M (mixture), and F (fruit). We can use the given information to fill in the Venn diagram step by step.

1. Start by writing the number of species that ate each food in the respective circles.

- S: 20 (species ate sunflower seeds)
- M: 22 (species ate the mixture)
- F: 11 (species ate the fruit)

2. Now fill in the overlapping regions based on the given information.

- 10 species ate both sunflower seeds and the mixture, so we write this number in the overlapping region between S and M.
- 4 species ate both sunflower seeds and the fruit, so we write this number in the overlapping region between S and F.
- 3 species ate both the mixture and the fruit, so we write this number in the overlapping region between M and F.
- 1 species ate all three foods, so we write this number in the very center of the Venn diagram where all three circles intersect.

Now let's calculate the answers to the specific questions:

A. None of the foods: To find the number of species that ate none of the foods, we need to subtract the total number of species (41) from the sum of all the individual foods and the overlapping regions.

None = Total species - S - M - F + (S ∩ M) + (S ∩ F) + (M ∩ F) - (S ∩ M ∩ F)

None = 41 - 20 - 22 - 11 + 10 + 4 + 3 - 1
None = 4

So, 4 species ate none of the foods.

B. The sunflower seeds, but neither of the other two foods: To find the number of species that only ate sunflower seeds, we need to subtract the overlapping regions that include S from the total number of species that ate S.

Sunflower seeds only = S - (S ∩ M) - (S ∩ F) - (S ∩ M ∩ F)

Sunflower seeds only = 20 - 10 - 4 - 1
Sunflower seeds only = 5

So, 5 species ate sunflower seeds only.

C. The mixture and the fruit, but not the sunflower seeds: To find the number of species that ate the mixture and the fruit only, we need to subtract the overlapping regions that include M and F from the total number of species that ate both the mixture and the fruit.

Mixture and fruit only = (M ∩ F) - (S ∩ M ∩ F)

Mixture and fruit only = 3 - 1
Mixture and fruit only = 2

So, 2 species ate the mixture and the fruit, but not the sunflower seeds.

D. The mixture or the fruit, but not the sunflower seeds: To find the number of species that ate either the mixture or the fruit, but not sunflower seeds, we need to consider the overlapping regions that include either M or F but not S.

Mixture or fruit only = (M ∩ ~S) + (F ∩ ~S) - (M ∩ F ∩ ~S)

Mixture or fruit only = (22 - 10) + (11 - 4) - 0
Mixture or fruit only = 19 + 7
Mixture or fruit only = 26

So, 26 species ate either the mixture or the fruit, but not the sunflower seeds.

E. Exactly one of the foods: To find the number of species that ate exactly one of the foods, we can sum the number of species that only ate sunflower seeds, only ate the mixture, or only ate the fruit.

Exactly one food = (Sunflower seeds only) + (Mixture only) + (Fruit only)

Exactly one food = 5 + (M - (M ∩ F) - (M ∩ S)) + (F - (F ∩ M) - (F ∩ S))
Exactly one food = 5 + (22 - 3 - 10) + (11 - 3 - 4)
Exactly one food = 5 + 9 + 4
Exactly one food = 18

So, 18 species ate exactly one of the foods.

In summary:
A. None of the foods: 4 species
B. The sunflower seeds, but neither of the other two foods: 5 species
C. The mixture and the fruit, but not the sunflower seeds: 2 species
D. The mixture or the fruit, but not the sunflower seeds: 26 species
E. Exactly one of the foods: 18 species