Six people are seated around a circular table. There are at least two men and two women. There are at least three right-handed persons. Every woman has a left-handed person to her immediate right. None of the women are right-handed. The number of women at the table is
A. 2
B. 3
C. 4
D. Cannot be determined
GATE 2017 · General Aptitude · Round Table Arrangement · medium
Answer: A. 2
Establish that all women are left-handed: The problem states none of the women are right-handed. Therefore every woman is left-handed. Consequently every right-handed person at the table must be a man.
Bound the number of women using the right-handed constraint: There are at least 3 right-handed people, and all right-handed people are men. So there are at least 3 men. With 6 seats total and at least 2 women, the number of women is between 2 and 3.
Test 3 women (3 men all right-handed): If there are 3 women and 3 men, and at least 3 right-handed persons (all men), then all 3 men are right-handed and there are no left-handed men. Each woman requires a left-handed person to her immediate right. That left-handed person can only be another woman (since no men are left-handed). So every woman must have a woman to her right. This forces all 3 women to sit consecutively in a block: W1-W2-W3-?. But W3 also needs a left-handed person to her right, which must be a woman - yet no fourth woman exists. Contradiction: 3 women is impossible.
Verify 2 women works: With 2 women and 4 men: at least 3 men are right-handed, so exactly 1 man can be left-handed. Arrangement example: W1 - LH_Man - M_R - W2 - LH_Man - M_R (but we only have 1 left-handed man). Try: W1(LH)-LH_Man-RH_Man-W2(LH)-LH_Man-RH_Man. With only 1 LH man: place the same LH man to the right of both women is impossible (he is one person). So we need at least 2 left-handed men. With 2 women needing 2 left-handed escorts and at least 3 right-handed men: 2LH men + 3RH men = 5 men total - but we only have 4 men. Try: if LH man serves one woman and the other woman has another woman to her right: W1-W2-LH_Man-RH-RH-RH. Check: W1 has W2 to the right (W2 is left-handed - valid). W2 has LH_Man to the right (valid). 3 right-handed men exist. 2 women >= 2. This arrangement satisfies all constraints.