ive the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure

U| = 150, |A| = 35, |B| = 12, |C| = 12 |A∩B| = 4, |A∩C| = 3, |B∩C| = 3, |A∩B∩C| = 1

To find the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, and VIII in the figure, we can use the principle of inclusion-exclusion.

The principle states that the size of the union of two sets A and B can be calculated as the sum of the sizes of each set minus the size of their intersection. We can extend this principle to three sets as well.

Let's break down the problem step by step using the given information:

1. Region I: This corresponds to the set A∩B∩C. We are given that |A∩B∩C| = 1. Therefore, the number of elements in region I is 1.

2. Region II: This corresponds to the set A∩B but excluding the elements in region I. To find the number of elements in region II, we need to subtract the elements in region I from |A∩B|. Given that |A∩B| = 4 and |A∩B∩C| = 1, we subtract 1 from 4 to get the number of elements in region II as 3.

3. Region III: This corresponds to the set A∩C but excluding the elements in region I. Similar to region II, we need to subtract the elements in region I from |A∩C|. Given that |A∩C| = 3 and |A∩B∩C| = 1, we subtract 1 from 3 to get the number of elements in region III as 2.

4. Region IV: This corresponds to the set B∩C but excluding the elements in region I. Similar to regions II and III, we need to subtract the elements in region I from |B∩C|. Given that |B∩C| = 3 and |A∩B∩C| = 1, we subtract 1 from 3 to get the number of elements in region IV as 2.

5. Region V: This corresponds to the set A but excluding the elements in regions I, II, and III. To find the number of elements in region V, we need to subtract the elements in regions I, II, and III from |A|. Given that |A| = 35, and we have already accounted for 1 + 3 + 2 elements in the previous regions, we subtract 6 from 35 to get the number of elements in region V as 29.

6. Region VI: This corresponds to the set B but excluding the elements in regions I, II, and IV. Similar to region V, we need to subtract the elements in regions I, II, and IV from |B|. Given that |B| = 12, and we have already accounted for 1 + 3 + 2 elements in the previous regions, we subtract 6 from 12 to get the number of elements in region VI as 6.

7. Region VII: This corresponds to the set C but excluding the elements in regions I, III, and IV. Similar to regions V and VI, we need to subtract the elements in regions I, III, and IV from |C|. Given that |C| = 12, and we have already accounted for 1 + 2 + 2 elements in the previous regions, we subtract 5 from 12 to get the number of elements in region VII as 7.

8. Region VIII: This corresponds to the universal set U excluding the elements in regions I, II, III, IV, V, VI, and VII. To find the number of elements in region VIII, we need to subtract the elements in regions I, II, III, IV, V, VI, and VII from |U|. Given that |U| = 150, and we have already accounted for all the elements in the previous regions, we subtract 1 + 3 + 2 + 2 + 29 + 6 + 7 elements from 150 to get the number of elements in region VIII as 100.

In summary, the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, and VIII are as follows:
I: 1
II: 3
III: 2
IV: 2
V: 29
VI: 6
VII: 7
VIII: 100