Answer to Question #318541 in Discrete Mathematics for eddie

Question

In a party, there are 12 people with white shirts and 7 people with red shirts;

4 people have black shoes and white shirts;

5 people have black shoes and red shirts;

the total number of people with white or red shirts or black shoes is 21.

How many people have black shoes?

Answer 1

Let

A - people with white shirts,

B - people with red shirts,

C - people with black shoes.

$|A\cup B\cup C|=|A|+|B|+|C|-\\-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|.$

Here $A\cup B\cup C-$ people with white or red shirts or black shoes, $|A\cup B\cup C|=21;$

$A\cap B-$ ​people with white and red shirts,$|A\cap B|=0$ (everybody wears one shirt) ;

$B\cap C-$ ​people with red shirts and black shoes,$|B\cap C|=5;$

$A\cap C-$ ​people with white shirts and black shoes,$|A\cap C|=4;$

$A\cap B\cap C-$ ​people with white and red shirts and black shoes,$|A\cap B\cap C|=0.$

So,

$21=12+7+|C|-0-5-4+0;$

$|C|=11-$ ​the number of people with black shoes.