Subjective Type

A group of $$123$$ workers went to a canteen for cold drinks, ice-cream and tea, $$42$$ workers took ice-cream, $$36$$ tea and $$30$$ cold drinks. $$15$$ workers purchase ice cream and tea, $$10$$ ice cream and cold drinks, and $$4$$ cold drinks and tea but not ice cream, $$11$$ took ice cream and tea but not cold drinks. Determine how many workers did not purchase anything ?

Solution

Given $$N=123$$, $$n(I)=42$$, $$n(T)=36$$,
$$n(C)=20$$, $$n(I\cap T)=15$$, $$n(I\cap C)=10$$,
$$n(C\cap T\cap I')=4$$, $$n(I\cap T\cap C')=11$$,
To find $$n(I'\cap T'\cap C')$$
We have
$$4=n(C\cap T\cap I')=n(C\cap T)-n(C\cap T\cap I')$$
and $$11=n(I\cap T\cap C')$$
$$=n(I\cap T)-n(I\cap T\cap C')$$
$$=15-n(I\cap T\cap C)$$
$$\therefore n(I\cap T\cap C)=4$$
Then $$(1)$$ Gives $$4=n(C\cap T)-4$$
or $$n(C\cap T)=8$$
Now $$n(I'\cap T'\cup C')=n(I\cup T\cup C)'$$
by De-Morgan law
$$=N-n(I\cup T\cup C)$$
$$=N-\left\{ n\left( I \right) +n\left( T \right) +n\left( C \right) -n\left( I\cap T \right) -n\left( T\cap C \right) -n\left( C\cap I \right) +n\left( I\cap T\cap C \right) \right\} $$
$$=123-\left\{ 42+36+30-15-8-10+4 \right\} =44$$

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