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Difficulty: MediumSets, Set Operations, and Venn Diagrams

In an agricultural survey of 100100 local farmers in a community, 5555 grow Cassava (CC), 4545 grow Yam (YY), and 4040 grow Maize (MM). It is observed that 2020 farmers grow both Cassava and Yam, 1515 grow both Yam and Maize, and 1616 grow both Cassava and Maize. If 77 farmers grow all three crops, how many farmers grow none of these three crops?

  1. 44Answer
  2. B
    1111
  3. C
    1616
  4. D
    2424

Answer

The number of farmers who grow none of the three crops is 4.
The number of farmers who grow at least one crop is found using n(C∪Y∪M)=55+45+40−20−15−16+7=96n(C \cup Y \cup M) = 55 + 45 + 40 - 20 - 15 - 16 + 7 = 96. Subtracting this union from the total number of farmers (100100) gives 100−96=4100 - 96 = 4 farmers who grow none of the three crops.

Step-by-Step Solution

1
Apply the Principle of Inclusion-Exclusion for three sets to find the total number of farmers growing at least one crop n(C∪Y∪M)n(C \cup Y \cup M).
n(C∪Y∪M)=n(C)+n(Y)+n(M)−n(C∩Y)−n(Y∩M)−n(C∩M)+n(C∩Y∩M)n(C \cup Y \cup M) = n(C) + n(Y) + n(M) - n(C \cap Y) - n(Y \cap M) - n(C \cap M) + n(C \cap Y \cap M)
Elements in pairwise intersections are double-counted and elements in all three sets are triple-counted, requiring correction.
2
Substitute the given cardinalities into the formula.
n(C∪Y∪M)=55+45+40−20−15−16+7=96n(C \cup Y \cup M) = 55 + 45 + 40 - 20 - 15 - 16 + 7 = 96
Calculating the total number of elements inside the union of the three sets.
3
Subtract the number of farmers in the union from the universal set size n(U)=100n(U) = 100.
n((C∪Y∪M)′)=100−96=4n((C \cup Y \cup M)') = 100 - 96 = 4
The complement of the union gives the number of elements belonging to none of the sets.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Universal Complement
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Topics

Sets, Set Operations, and Venn Diagrams

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