Answer to Question #286850 in Discrete Mathematics for Pearls

Question

Each of 39 of my friends has either a dog a cat or a rabbit each of 24 of them has a dog each of 17 gas a cat and each of 16 has a rabbit the number having both a dog and a cat is one more than the number having both a cat and a rabbit while 2of them have all the three how many of my friends have both a rabbit and cat

Answer 1

Each of 39 of my friends has either a dog, a cat or a rabbit. Each of 24 of them has a dog, each of 17 has a cat and each of 16 has a rabbit. The number having both a dog and a cat is 1 more than the number having both a cat and a rabbit. There are 9 who have both a dog and a rabbit while 2 of them have all the three. How many of my friends have both a rabbit and a cat?

Let $x=$ the number having both a cat and a rabbit. Then $x+1=$ the number having both a dog and a cat.

$$N(D\cup C\cup R)=N(D)+N(C)+N(R)$$

$$-N(D\cap C)-N(D\cap R)-N(C\cap R)$$

$$+N(D\cap C\cap R)$$

$$39=24+17+16-(x+1)-x-9+2$$

$$2x=10$$

$$x=5$$

There are 5 who have both a rabbit and a cat.