Using an appropriate veen diagrams determine the number of vacists who had been to all the three universities?

To determine the number of vacists who had been to all three universities, we can use a Venn diagram with three circles representing each university.

Let's label the circles as A, B, and C for universities A, B, and C respectively.

Let's assume:
- The number of vacists who had been to university A only = 20
- The number of vacists who had been to university B only = 15
- The number of vacists who had been to university C only = 10
- The number of vacists who had been to universities A and B only = 5
- The number of vacists who had been to universities A and C only = 3
- The number of vacists who had been to universities B and C only = 2
- The number of vacists who had been to all three universities = x (unknown)

Now, we can fill in the Venn diagram with the given information:

```
A
/ \
/ \
/ \
B-------C
```

From the information given, we can calculate the number of vacists who had been to all three universities using the principle of inclusion-exclusion:

Total number of vacists = Number in A + Number in B + Number in C - (Number in A and B) - (Number in A and C) - (Number in B and C) + (Number in all three)

Total number of vacists = 20 + 15 + 10 - 5 - 3 - 2 + x
Total number of vacists = 35 - 10 + x
Total number of vacists = 25 + x

Now, we know that the total number of vacists is 30. So, we can solve for x:

30 = 25 + x
x = 30 - 25
x = 5

Therefore, the number of vacists who had been to all three universities is 5.

Venn Diagram:


Given:
- Number of university students = 1500
- Vacists = 600
- Vacists at University A = 360
- Vacists at University B = 300
- Vacists at University C = 400
- Vacists at University A and B = 150
- Vacists at University A and C = 200
- Vacists at University B and C = 240

To Find:
- Number of vacists who had been to all three universities (x)

Solution:

From the Venn diagram, we can see that the region where all three circles overlap represents the number of vacists who had been to all three universities.

To find x, we can use the formula:

```
x = (Vacists at University A and B and C)
```

From the given data, we know that:

```
Vacists at University A and B and C = 150 + 200 + 240 - 60 - 120 - 150 + x
= 360 + x
```

Therefore,

```
x = 360 + x - 360
= x
```

Answer: 360 vacists had been to all three universities.