SOLUTION: There is a group of 85 teachers. 40 of them have CPR training, 33 have fitness training and 42 have water safety training. 15 have CPR and fitness training, 12 have CPR and water

Question 1155285: There is a group of 85 teachers. 40 of them have CPR training, 33 have fitness training and 42 have water safety training. 15 have CPR and fitness training, 12 have CPR and water training, 10 have fitness training and water safety training and 5 have all three types of training.
a) Make a three circle Venn diagram to illustrate these results.
b) How many teachers have exactly two of these training qualifications?

Answer by ikleyn(52894) (Show Source): You can put this solution on YOUR website!
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This problem in its part (b) can be easily solved WITHOUT drawing the Venn diagram / circles.

    The number of those who is trained in (CRR & fitness)   ONLY  is  15-5 = 10.


    The number of those who is trained in (CRR & water)     ONLY  is  12-5 = 7.


    The number of those who is trained in (fitness & water) ONLY  is  10-5 = 5.


    These three listed subsets are DISJOINT (!)  ---- therefore, the answer to the problem's question is  10 + 7 + 5 = 22.

Solved.

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In many combinatoric problems the pure logic approach works better than Venn diagrams and (in my opinion) is much more educative.

The given problem belongs exactly to this category.

I admit that other people may have different view.

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