The 130 consumers were asked about their buying preferences concerning another product that is sold in the market under three labels. The results were
12 buy only those sold under label A.
24 buy only those sold under label B.
27 buy only those sold under label C.
14 buy only those sold under labels A and B.
9 buy only those sold under labels A and C.
11 buy only those sold under labels B and C.
7 buy the product sold under all three labels.
How many of the consumers surveyed buy the product sold under none of these labels?
a. 106
b. 89
c. 41
d. 24
e. 19
This is confusing. Please explain. Thank you
To find the number of consumers who buy the product sold under none of the labels, we can use the principle of inclusion-exclusion. The principle of inclusion-exclusion states that to find the total count of a set of items, we need to consider the individual counts of each subset and then subtract the counts of their intersections.
In this case, we have three subsets: consumers who buy the product under label A, consumers who buy under label B, and consumers who buy under label C. We also have the counts of the intersections between these subsets.
Let's break it down step by step:
1. Start with the total count of consumers surveyed, which is given as 130.
Total count = 130
2. Now, subtract the counts of consumers in each individual subset.
Consumers who buy under label A = 12
Consumers who buy under label B = 24
Consumers who buy under label C = 27
Total count - Consumers who buy under label A - Consumers who buy under label B - Consumers who buy under label C
= 130 - 12 - 24 - 27
3. Next, add back the counts of intersections between subsets.
Consumers who buy under labels A and B = 14
Consumers who buy under labels A and C = 9
Consumers who buy under labels B and C = 11
Total count - Consumers who buy under label A - Consumers who buy under label B - Consumers who buy under label C
+ Consumers who buy under labels A and B + Consumers who buy under labels A and C + Consumers who buy under labels B and C
= 130 - 12 - 24 - 27 + 14 + 9 + 11
4. Finally, subtract the count of consumers who buy the product under all three labels.
Consumers who buy under all three labels = 7
Total count - Consumers who buy under label A - Consumers who buy under label B - Consumers who buy under label C
+ Consumers who buy under labels A and B + Consumers who buy under labels A and C + Consumers who buy under labels B and C
- Consumers who buy under all three labels
= 130 - 12 - 24 - 27 + 14 + 9 + 11 - 7
This calculation will give us the number of consumers who buy the product sold under none of the labels.