GATE EE 2017 Set 1 — Question 62

Let $W$ be the number of women and $M$ be the number of men. We are given:From (3) and (4), since all women are LH, $W \leq 3$. Thus, $W$ can be 2 or 3.Case 1: $W = 3$. All 3 women are LH. To satisfy condition (5), each woman must have a LH person to her right. If they sit in a chain $W_1 \to W_2 \to W_3$, then $W_3$ must have another LH person to her right. Since all women are already accounted for, this person must be a LH man. This would mean there are 4 LH people (3 women + 1 man), which contradicts the condition that there are at most 3 LH people. If they don't sit together, say $W_1 \to M_1 \to W_2 \to M_2 \dots$, then every man to the right of a woman must be LH, leading to even more LH people. Thus, $W=3$ is impossible.Case 2: $W = 2$. Both women are LH. They can satisfy condition (5) if they sit as $W_1 \to W_2 \to M_{LH}$, where $M_{LH}$ is a LH man. This configuration uses 3 LH people (2 women + 1 man) and 3 RH people (the remaining men), which satisfies all given conditions. Therefore, the number of women is 2.