Problem 11 · Swimming, football and handball
Inclusion–exclusion with three sets.
Integer answer, at most 4 digitsFifty students at a school are surveyed: 8 do swimming, football and handball; 19 do swimming and football; 17 swimming and handball; 20 football and handball; 28 swimming; 34 football; and 31 handball. How many of the students do none of these sports?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: inclusion–exclusion: add the three sets, subtract the pairwise intersections and add the triple one back.
$$|N \cup F \cup H| = 28 + 34 + 31 - 19 - 17 - 20 + 8 = 45.$$
$$50 - 45 = 5\ \text{students do none of these sports}.$$
Answer: 5