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.

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.

"At least one salesman" means visited by A only, by B only, or by both — in set terms it is the union \(n(A \cup B)\).

The information given states that out of the 20 villages, none of them visited 2 villages. Those 2 are exactly the villages lying outside the union.

So the union is the remainder: \(n(A \cup B) = 20 - 2 = 18\). The answer follows straight from the total without needing the individual figures 14 and 12 at all.

A useful check is that the union can never exceed the 20 villages available, which is why simply adding 14 and 12 to get 26 cannot be a count of villages — that sum double-counts the villages both salesmen reached.

Hence 18 villages were visited by at least one salesman.