In a gym attended by 95 people, 35 people attend the judo course, 42 the karate course and 22 the yoga course. Knowing that: 2 people attend all three courses, 8 attend both judo and yoga, 6 attend both yoga and karate and 20 attend only karate lessons, how many people do not attend any course?

A: 8

B: 26

C: 21

D: 16

E: 12

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Hi!

This is the best i could do, is the answer 26?

I’m not sure why we would need to subtract the 2 people who attend all three courses?

This problem can be solved using the principle of inclusion-exclusion, which is a counting technique used to determine the number of elements in the union of two or more sets. In this case, we have three sets: the set of people attending the judo course, the set of people attending the karate course, and the set of people attending the yoga course.

Let’s call the set of people attending the judo course J, the set of people attending the karate course K, and the set of people attending the yoga course Y.

First, we find the total number of people attending at least one course by finding the union of the three sets:

|J ∪ K ∪ Y| = |J| + |K| + |Y| - |J ∩ K| - |J ∩ Y| - |K ∩ Y| + |J ∩ K ∩ Y|

Where |S| represents the number of elements in set S.

We are given the following information:

|J| = 35

|K| = 42

|Y| = 22

|J ∩ Y| = 8

|K ∩ Y| = 6

|J ∩ K| = 2 people attend both judo and karate lessons (this information is not given, so we have to find it)

|J ∩ K ∩ Y| = 2 people attend all three courses

To find |J ∩ K|, we subtract the number of people attending only karate lessons (20) from the total number of people attending karate and judo lessons (42):

|J ∩ K| = 42 - 20 = 22

Now that we have all the information, we can plug it into the formula:

|J ∪ K ∪ Y| = 35 + 42 + 22 - 22 - 8 - 6 + 2 = 75

Finally, to find the number of people who do not attend any course, we subtract the total number of people attending at least one course from the total number of people in the gym:

95 - 75 = 20

So, the answer is 20, or option B: 26 people do not attend any course.

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