Question #318541

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?

1
Expert's answer
2022-03-29T01:11:43-0400

Let

A - people with white shirts,

B - people with red shirts,

C - people with black shoes.


ABC=A+B+CABBCAC+ABC.|A\cup B\cup C|=|A|+|B|+|C|-\\-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|.


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


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

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


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


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


So,

21=12+7+C054+0;21=12+7+|C|-0-5-4+0;

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


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