Sample probability question from an actuarial test

Please help me understand this question.

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A survey of a group’s viewing habits over the last year revealed the following

information:

(i) 28% watched gymnastics

(ii) 29% watched baseball

(iii) 19% watched soccer

(iv) 14% watched gymnastics and baseball

(v) 12% watched baseball and soccer

(vi) 10% watched gymnastics and soccer

(vii) 8% watched all three sports.

Calculate the percentage of the group that watched none of the three sports

during the last year.

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The way I approached the question is like this.

Let

P(i) = 28%

P(ii) = 29%

P(iii) = 19%

P(iv) = 14%

P(v) = 12%

P(vi) = 10%

P(vii) = 8%

Obviously, the answer isn't simply 100% - P(i) - P(ii) - P(iii) because there are bunch of "overlaps" going on. To compensate for these "overlaps", you need to add back P(iv), P(v), and P(vi). But this would be adding back too much. So, you need to take away a certain multiple of P(vii). I thought you needed to take away 3 * P(vii), but the answer says take away 1 * P(vii).

If you followed what I said, I thought the answer would be

100% - P(i) - P(ii) - P(iii) + P(iv) + P(v) + P(vi) - P(vii) - P(vii) - P(vii) = 36

But the correct answer is

100% - P(i) - P(ii) - P(iii) + P(iv) + P(v) + P(vi) - P(vii) = 52

Can someone explain this?

(I wish I can draw a picture on here so I can explain what I'm thinking/visualizing. When I did this, I imagined 3 colored circles overlapping and the areas I would take away and add back to get an area where there wouldn't be any overlaps. When I visualized the problem in this way I thought the answer I came up with was correct, but it wasn't)