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  4. Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey; 80 played cricket and basketball and 40 played cricket and hockey; 24 played all the three games. The number of boys who did not play any game is

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Q. Out of $ 800 $ boys in a school, $ 224 $ played cricket, $ 240 $ played hockey and $ 336 $ played basketball. Of the total, $ 64 $ played both basketball and hockey; $ 80 $ played cricket and basketball and $ 40 $ played cricket and hockey; $ 24 $ played all the three games. The number of boys who did not play any game is

UPSEEUPSEE 2012

Solution:

Given, $n(U)=800$,
$n(C)=224, n(H)=240, n(B)=336$
$n(H \cap B)=64, n(B \cap C)=80$
$n(H \cap C)=40, n(C \cap H \cap B)=24$
$\therefore n\left(C^{c} \cap H^{c} \cap B^{c}\right)=n\left[(C \cup H \cup B)^{c}\right]$
$=n(U)-n(C \cup H \cup B)$
$=800-[n(C)+n(H)+n(B)-n(H \cap C)$
$-n(H \cap B)-n(C \cap B)+n(C \cap H \cap B)]$
$=800-[224+240+336-40-64-80+24]$
$=800-640=160$