Answer to Question #327664 in Discrete Mathematics for Sammy

Question

    In a group of 35 ex-athletes, 17 play golf, 20 go cycling, and 12 do yoga. Exactly 8 play golf and go cycling, 8 play golf and do yoga, 7 go cycling and do yoga, and 4 do all three activities. How many of the ex-athletes do none of these activities? 

Answer 1

$A-golf\\B-cycling\\C-yoga\\|\bar{A}\cap \bar{B}\cap \bar{C}|=|\varOmega |-\left| A\cup B\cup C \right|=\\=35-\left( \left| A \right|+\left| B \right|+\left| C \right|-\left| A\cap B \right|-\left| A\cap C \right|-\left| B\cap C \right|+\left| A\cap B\cap C \right| \right) =\\=35-\left( 17+20+12-8-8-7+4 \right) =5$