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Question:

Based on the information given above, the minimum number of volunteers involved in both FR and TR projects, but  not in the ER project is:

Explanation:

17 volunteers are involved in the TR project and 10 in TR are also involved in other projects. Thus, 7 volunteers are involved only in TR.

∴ 8 volunteers are involved in ER alone.

∴ 4 volunteers are involved in all the three projects.

Let x people be involved in FR alone.

∴ Number of people involved in FR and ER but not TR = x – 4

Now, a + b + 4 = 10 

∴ a + b = 6

Also, 7 + a + b + 4 + x + x – 4 + 8 = 37

∴ 2x = 16 or x = 8 

Number of Volunteers involved in FR > Number of Volunteers involved in TR

And Number of Volunteers involved in FR > Number of Volunteers involved in ER

∴ 16 + a > 17 and 16 + a > 16 + b or a > b

∴ a and b can be (6, 0), (5, 1), (4, 2)

The minimum number of volunteers involved in both FR and TR projects, but not in the ER Project = minimum value of a = 4

Hence, option (c).