Each of the question below consists of a question and three statements number I, II and III given below it. You have to decide whether the data provided in the statement are sufficient to answer the question. A bag contains some number of Diamond of 4 different colors (Red, Pink, blue and yellow). What is the total number of Diamond in the bag? I. If 5 Pink Diamond from the bag are replaced by Yellow Diamond, Then the ratio of probabilities of picking 2 yellow Diamond at a time to picking 3 Blue Diamond at a time becomes (1/116). II. The different between probabilities of picking 2 Red Diamonds at a time and probability of picking 2 Blue Diamonds at a time is (23/330). III. If a Diamond is picked from the bag, then there is 40% chance that it is a red Diamond. Probability of picking 2 Red Diamond from the bag is (26/165). Correct Answer The data in only statement III alone is sufficient to answer the question. From Statement I: Let number of Red, Pink, Blue and Yellow Diamond in the bag be P, Q, R and S respectively. Let total number of diamond in the bag be T T = (P + Q + R + S) After replacing 5 Pink Diamond by Yellow diamond. Probability of picking 2 Yellow diamond = ((S + 5 )C2)/ (TC2) = ((S + 5) × (S + 4))/ (T × (T – 1)) Probability of packing 3 Blue diamond = ((R)C2)/ (TC2) = (R × (R - 1) × (R - 2))/ (T × (T – 1) × (T – 2)) The ratio of probabilities of picking 2 Yellow diamond at a time to picking 3 Blue diamond at a time = (((S + 5) × (S + 4))/ (T × (T - 1)))/( (R × (R – 1) × (R – 2))/(T × (T – 1) × (T – 2))) = (((S + 5) × (S + 4) × (T – 2))/(R × (R – 1) × (R – 2))) = (1/116) ⇒ 116 × (S + 5) × (S + 4) × ((T – 2) = (R × (R – 1) × (R – 2)) ⇒ 116 × (S + 5) × (S + 4) × ((T – 2) = (R × (R – 1) × (R – 2)) So, data in statement I alone is not sufficient to answer the question. From Statement II: Let number of Red diamonds and number of Blue diamonds in the bag be P and R respectively. Lets total number of diamond be T Probability of picking 2 Red diamonds = (pC2)/ (TC2) = (P × (P – 1))/ (T × (T – 1)) Probability of picking 2 Blue diamonds = (RC2)/ (TC2) = (R × (R – 1))/ (T × (T – 1)) As per given, ((P × (P – 1))/ (T × (T – 1))) – ((R × (R – 1)) / (T × (T – 1))) = (23/330) ((P × (P – 1)) - (R × (R – 1))) = (23/330) × (T × (T – 1)) So, data in statement II alone is not sufficient to answer the question. Even combining data from statement I and II together is also not sufficient to answer the question. From Statement III: Probability of picking a red diamond = (Number of red diamond)/ (Total number of diamond) = (40/100) = (2/5) So, let number of red diamond be 2T and total number of diamond be 5T Probability of picking 2 red diamond from the bag = (2TC2)/(5TC2) = (2T × (2T – 1))/ (5T × (5T -1)) = (26/165) (4T2 – 2T)/ (25T2 – 5T) = (26/165) 660T2 – 330T = 650T2 – 130T 10T2 = 200T T = 20 Total number of diamonds in the bag is 100. Data in statement III alone is sufficient to answer the question.