In a certain zoo, there are 42 animals in one sector, 34 in the second sector and 20 in the third sector. Out ofthis, 24 graze in sector one and also in sector two. 10 graze in sector two and sector three. 12 graze in sector one and sector three. These figures also include four animals grazing in all the three sectors. If all these animals are now transported to another zoo, then find the total number of animals?

Options

A. 38
B. 56
C. 54
D. None of these

Answer

To find the total number of animals in the zoo, we will use the principle of inclusion-exclusion. Let's denote the sectors as follows:

- Sector A (first sector) has 42 animals.
- Sector B (second sector) has 34 animals.
- Sector C (third sector) has 20 animals.

We also have the following information about the animals grazing in multiple sectors:

- 24 animals graze in both Sector A and Sector B (N(A ∩ B) = 24).
- 10 animals graze in both Sector B and Sector C (N(B ∩ C) = 10).
- 12 animals graze in both Sector A and Sector C (N(C ∩ A) = 12).
- 4 animals graze in all three sectors (N(A ∩ B ∩ C) = 4).

Now we can apply the formula for the union of three sets:

\[ N(A \cup B \cup C) = N(A) + N(B) + N(C) - N(A \cap B) - N(B \cap C) - N(C \cap A) + N(A \cap B \cap C) \]

Substituting the values we have:

\[ N(A \cup B \cup C) = 42 + 34 + 20 - 24 - 10 - 12 + 4 \]

Now, let's calculate step by step:

1. **Calculate the sum of animals in each sector:**
\[ 42 + 34 + 20 = 96 \]

2. **Calculate the sum of animals grazing in two sectors:**
\[ 24 + 10 + 12 = 46 \]

3. **Now substitute these values into the formula:**
\[ N(A \cup B \cup C) = 96 - 46 + 4 \]

4. **Calculate the total:**
\[ 96 - 46 = 50 \]
\[ 50 + 4 = 54 \]

Thus, the total number of animals transported to another zoo is **54**.