Dogs in the GoodDog Obedience School win a blue ribbon for learning how to sit, a green ribbon for learning how to roll over, and a white ribbon for learning how to stay. There are 100 dogs in the school. 62 have blue ribbons, 55 have green ribbons, and 63 have white ribbons.

32 have a blue ribbon and a green ribbon 31 have a green ribbon and a white ribbon; 38 have a blue ribbon and a white ribbon.

16 have all three ribbons.

How many dogs have not learned any tricks?

  5

Step-by-step explanation:

The 32 that have blue and green ribbons include the 16 that have all three, so there are only 32 -16 = 16 that have only blue and green ribbons.

The 31 that have green and white ribbons likewise include the 16 with all three, so there are only 31 -16 = 15 that have only green and white ribbons.

The 38 that have blue and white ribbons include the 16 with all three, so there are only 38 -16 = 22 that have only green and white ribbons.

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If we add the numbers of blue, green, and white ribbons, we are counting twice the numbers that have 2 ribbons, and 3 times the numbers that have 3 ribbons. We want to count each kind of ribbon-holder only once. Hence the number of individual dogs with any number of ribbons is only ...

  62 +55 +63 -(16 +15 +22) -2(16) = 95

Of the 100 dogs, 95 have ribbons, so 5 dogs have not learned any tricks.

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