Let be a complex cube root of unity with 1. A fair die is thrown three times. If r_1, r_2 and r_3 are the numbers obtained on the die, then the probability that ^ r_1 + ^ r_2 + ^ r_3 = 0 is

JEE Advanced 2010 Mathematics question

Let be a complex cube root of unity with . A fair die is thrown three times. If , and are the numbers obtained on the die, then the probability that is

(A)
(B)
(C)Correct
(D)

Answer: (C)

Solution

Each equals , or according as . A sum of three such numbers is zero only when it is , i.e. when leave three different remainders modulo 3 (if two or three of them are equal the sum is non-zero, e.g. $1+1+ω = 1

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Question ID VK-025764