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?
A−golfB−cyclingC−yoga∣Ω∣=35∣A∣=17,∣B∣=20,∣C∣=12∣A∩B∣=8,∣A∩C∣=8,∣B∩C∣=7∣A∩B∩C∣=4∣Aˉ∩Bˉ∩Cˉ∣=∣Ω∣−∣A∪B∪C∣==∣Ω∣−(∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣)=5=35−(17+20+12−8−8−7+4)=5A-golf\\B-cycling\\C-yoga\\\left| \varOmega \right|=35\\\left| A \right|=17,\left| B \right|=20,\left| C \right|=12\\\left| A\cap B \right|=8,\left| A\cap C \right|=8,\left| B\cap C \right|=7\\\left| A\cap B\cap C \right|=4\\\left| \bar{A}\cap \bar{B}\cap \bar{C} \right|=\left| \varOmega \right|-\left| A\cup B\cup C \right|=\\=\left| \varOmega \right|-\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) =5\\=35-\left( 17+20+12-8-8-7+4 \right) =5A−golfB−cyclingC−yoga∣Ω∣=35∣A∣=17,∣B∣=20,∣C∣=12∣A∩B∣=8,∣A∩C∣=8,∣B∩C∣=7∣A∩B∩C∣=4∣∣Aˉ∩Bˉ∩Cˉ∣∣=∣Ω∣−∣A∪B∪C∣==∣Ω∣−(∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣)=5=35−(17+20+12−8−8−7+4)=5
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