Question

a coffee shop collected the following information regarding purchases from 120 of its customers. 59 purchased coffee. 40 purchased donuts. 19 purchased coffee and donuts. complete parts a) through c). a) of those surveyed, how many purchased only coffee? 40 (type a whole number.) b) of those surveyed, how many purchased only donuts? 21 (type a whole number.) c) of those surveyed, how many did not purchase either of these items? (type a whole number.)

Explanation:

Step1: Calculate number of only - coffee buyers

The number of customers who bought coffee is 59, and 19 bought both coffee and donuts. To find the number of those who bought only coffee, we subtract the number of those who bought both from the total number of coffee - buyers. So, the number of only - coffee buyers is $59 - 19=40$.

Step2: Calculate number of only - donut buyers

The number of customers who bought donuts is 40, and 19 bought both coffee and donuts. To find the number of those who bought only donuts, we subtract the number of those who bought both from the total number of donut - buyers. So, the number of only - donut buyers is $40 - 19 = 21$.

Step3: Calculate number of customers who bought either coffee or donuts or both

We use the principle of inclusion - exclusion. The number of customers who bought either coffee or donuts or both is $n(C\cup D)=n(C)+n(D)-n(C\cap D)$, where $n(C) = 59$ (coffee buyers), $n(D)=40$ (donut buyers) and $n(C\cap D)=19$ (both buyers). So $n(C\cup D)=59 + 40-19=80$.

Step4: Calculate number of customers who bought neither

The total number of surveyed customers is 120. The number of customers who bought neither coffee nor donuts is $120-(59 + 40 - 19)=120 - 80=40$.

Answer:

40