Directions : Question numbers 31 and 32 are based on the following information : Out of 20 villages in a certain state, salesman A visited 14 villages, salesman B visited 12 villages. None of them visited 2 villages.
How many villages were visited by both the salesmen ?
8
Start from the total and remove the villages nobody reached, to find how many were visited by at least one salesman.
\(n(A \cup B) = 20 - 2 = 18\) villages were visited by A, by B, or by both.
Now apply the inclusion-exclusion principle, \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\). Simply adding 14 and 12 counts every village visited by both men twice, so the overlap has to be subtracted once to correct that double count.
Substituting the values gives \(18 = 14 + 12 - n(A \cap B)\), so \(n(A \cap B) = 26 - 18 = 8\).
The result is consistent, since 8 villages were visited by both, \(14 - 8 = 6\) by A alone and \(12 - 8 = 4\) by B alone, and \(6 + 8 + 4 + 2 = 20\) accounts for every village exactly once.
Hence 8 villages were visited by both the salesmen.
Similar Questions
Directions : Question numbers 31 and 32 are based on the following information :
Out of 20 villages in a certain state, salesman A visited 14 villages, salesman B visited 12 villages. None of them visited 2 villages.
How many villages were visited by at least one salesman ?
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