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

Answer
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Hint: We will proceed in this problem by making a venn diagram of the problem.
Let consider three sets i.e., C, I and T which represents the workers purchasing coffee, ice-cream and tea respectively.

Complete step-by-step answer:

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Total number of workers=123
Number of workers purchasing ice-cream, n(I)=42
Number of workers purchasing tea, n(T)=36
Number of workers purchasing coffee, n(C)=30
Number of workers purchasing ice-cream and tea, n(IT)=15
Number of workers purchasing ice-cream and coffee, n(IC)=10
Number of workers purchasing only ice-cream and tea but not coffee (shown in the figure through blue coloured hatched lines) is given by
n(IT)n(ITC)=1115n(ITC)=11n(ITC)=1511=4
Number of workers purchasing only coffee and tea but not ice-cream (shown in the figure through green coloured hatched lines) is given by
n(TC)n(ITC)=4n(TC)4=4n(TC)=8
 As we know that for any three sets i.e., C, I and T, we can write
n(ITC)=n(I)+n(T)+n(C)n(IT)n(TC)n(IC)+n(ITC) (1)
Now substituting all the values in equation (1), we get
Number of workers purchasing either ice-cream or tea or coffee is given by
n(ITC)=42+36+3015810+4=79
Since, Number of workers who did not purchase anything is equal to the total number of workers minus the number of workers purchasing either ice-cream or tea or coffee.
Number of workers who did not purchase anything=12379=44.

Note: In these types of problems, a venn diagram is used to calculate all the unknowns. In this particular problem, we used the given data to determine the unknowns in equation (1).
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