A group of market women sell at least one of yam, plantain, and maize. 12 of them sell maize, 10 sell yam, and 14 sell plantain. 5 sell plantain and maize, 4 sell yam and maize, 2 sell yam and plantain only, while 3 sell all three items. How many women are in the group?
A. 25
B. 19
C. 18
D. 17
Answer (2)
Define sets for women selling maize (M), yam (Y), and plantain (P).
Use the Principle of Inclusion-Exclusion: ∣ M c u p Y c u pP ∣ = ∣ M ∣ + ∣ Y ∣ + ∣ P ∣ − ∣ M c a p Y ∣ − ∣ M c a pP ∣ − ∣ Y c a pP ∣ + ∣ M c a p Y c a pP ∣ .
Calculate ∣ Y c a pP ∣ = 2 + 3 = 5 .
Substitute values: $|M cup Y cup P| = 12 + 10 + 14 - 4 - 5 - 5 + 3 =
\boxed{25}$.
Explanation
Analyze the problem Let's analyze the problem. We have information about the number of market women selling different combinations of yam, plantain, and maize. We need to find the total number of women in the group. We will use the Principle of Inclusion-Exclusion to solve this problem.
Define sets and given values Let M be the set of women who sell maize, Y be the set of women who sell yam, and P be the set of women who sell plantain.
We are given: ∣ M ∣ = 12 (number of women who sell maize) ∣ Y ∣ = 10 (number of women who sell yam) ∣ P ∣ = 14 (number of women who sell plantain) ∣ P c a pM ∣ = 5 (number of women who sell plantain and maize) ∣ Y c a pM ∣ = 4 (number of women who sell yam and maize) Number of women who sell yam and plantain only = 2, which means ∣ Y c a pP c a p M c ∣ = 2 Number of women who sell all three items = 3, which means ∣ Y c a pP c a pM ∣ = 3
Apply Principle of Inclusion-Exclusion We want to find the total number of women, which is ∣ M c u p Y c u pP ∣ .
Using the Principle of Inclusion-Exclusion: ∣ M c u p Y c u pP ∣ = ∣ M ∣ + ∣ Y ∣ + ∣ P ∣ − ∣ M c a p Y ∣ − ∣ M c a pP ∣ − ∣ Y c a pP ∣ + ∣ M c a p Y c a pP ∣
Calculate |Y ∩ P| We know that the number of women who sell yam and plantain only is 2, and the number of women who sell all three items is 3. Therefore, the total number of women who sell yam and plantain is: ∣ Y c a pP ∣ = ∣ Y c a pP c a p M c ∣ + ∣ Y c a pP c a pM ∣ = 2 + 3 = 5
Substitute values into the formula Now, we substitute all the known values into the Inclusion-Exclusion formula: ∣ M c u p Y c u pP ∣ = 12 + 10 + 14 − 4 − 5 − 5 + 3
Calculate the total number of women Calculating the result: ∣ M c u p Y c u pP ∣ = 12 + 10 + 14 − 4 − 5 − 5 + 3 = 36 − 14 + 3 = 22 + 3 = 25
Therefore, there are 25 women in the group.
State the final answer The total number of women in the group is 25.
Examples
This problem demonstrates how the Principle of Inclusion-Exclusion can be used in real-world scenarios, such as market research or surveys. For example, if a company wants to know how many customers like at least one of their three products, they can use this principle to avoid double-counting customers who like multiple products. By gathering data on the number of customers who like each product individually, the number who like each pair of products, and the number who like all three, they can accurately calculate the total number of customers who like at least one product. This helps the company understand the overall popularity of their product line and make informed business decisions.
By applying the Principle of Inclusion-Exclusion to the given sets of women selling maize, yam, and plantain, we calculated the total number of unique women in the group as 25. Therefore, the answer is option A: 25.
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