In a survey of 100 agricultural market traders in Lagos, 52 sell cassava, 45 sell yam, and… — JAMB UTME

In a survey of $100$ agricultural market traders in Lagos, $52$ sell cassava, $45$ sell yam, and $60$ sell plantain. Furthermore, $25$ sell both cassava and yam, $22$ sell both yam and plantain, and $28$ sell both cassava and plantain. If the number of traders who sell none of these three crops is twice the number of traders who sell all three crops, how many traders sell exactly two of these crops?

Answer: 57 traders

Answer

57 traders sell exactly two of these crops.

Using inclusion-exclusion, the total number of traders selling at least one crop is

82 + x

. Adding the

2x

traders selling none gives

82 + 3x = 100

, so

x = 6

. The number of traders selling exactly two crops is

(25 - 6) + (22 - 6) + (28 - 6) = 19 + 16 + 22 = 57

Step-by-Step Solution

Apply the Principle of Inclusion-Exclusion for three set unions.

n(C \cup Y \cup P) = 82 + x

, where

x = n(C \cap Y \cap P)

Summing single set cardinalities, subtracting pairwise intersections, and adding back the triple intersection accounts for all region overlaps.

Set up and solve the universal set cardinality equation.

x = 6

Since total traders

|U| = 100

and non-sellers equal

2x

, the equation

100 = (82 + x) + 2x

simplifies to

3x = 18

, giving

x = 6

Compute the sum of elements in regions representing exactly two sets.

Subtracting

x = 6

from each pairwise intersection isolates traders who sell only cassava & yam (19), only yam & plantain (16), and only cassava & plantain (22). Summing these gives

19 + 16 + 22 = 57

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Partitioning Venn Diagram Regions

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