Answer to Question #318539 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

$W-white\,\,shirts\\R-red\,\,shirts\\B-black\,\,shoes\\\left| W \right|=12,\left| R \right|=7,\left| W\cap R \right|=0,\left| W\cap B \right|=4,\left| R\cap B \right|=5,\left| W\cap R\cap B \right|=0,\left| W\cup R\cup B \right|=21\\Inclusion-exclusion\,\,principle:\\\left| W\cup R\cup B \right|=\left| W \right|+\left| R \right|+\left| B \right|-\left| W\cap R \right|-\left| W\cap B \right|-\left| R\cap B \right|+\left| W\cap R\cap B \right|\Rightarrow \\\Rightarrow \left| B \right|=21-12-7+4+5=11$