Answer to Question #315814 in Discrete Mathematics for nics

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 - the multiplicity of people with white shirts,

B - the multiplicity of people with red shirts,

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

$A\cap B-$ the multiplicity of people with white and red shirts, $|A\cap B|=0$ (we will assume that everybody wears one shirt;

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

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

$A\cap B\cap C-$ the multiplicity of 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.