Let omega be a complex cube root of unity wit | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The condition $\omega^{r_1} + \omega^{r_2} + \omega^{r_3} = 0$ is satisfied if the remainders of $r_1, r_2, r_3$ when divided by $3$ are ${0, 1, 2

Vedclass · Accepted Answer

$2/9$

Vedclass · Answer

(C) The condition $\omega^{r_1} + \omega^{r_2} + \omega^{r_3} = 0$ is satisfied if the remainders of $r_1, r_2, r_3$ when divided by $3$ are ${0, 1, 2}$ in some order.For a die,the possible values are ${1, 2, 3, 4, 5, 6}$.Remainders modulo $3$ are:$1 \equiv 1, 2 \equiv 2, 3 \equiv 0, 4 \equiv 1, 5 \equiv 2, 6 \equiv 0$.There are $2$ numbers for each remainder: $0$ (from ${3, 6}$),$1$ (from ${1, 4}$),and $2$ (from ${2, 5}$).Number of favorable outcomes $= (2 	imes 2 	imes 2) 	imes 3! = 8 	imes 6 = 48$.Total outcomes $= 6^3 = 216$.Probability $= \frac{48}{216} = \frac{2}{9}$.

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2,$ and $r_3$ are the numbers obtained on the die,what is the probability that $\omega^{r_1} + \omega^{r_2} + \omega^{r_3} = 0$?

Similar Questions

$A$ dice is rolled three times. The probability of getting a larger number than the previous number each time is:

$A$ mapping is selected at random from the set of all the mappings of the set $A = \{1, 2, ..., n\}$ into itself. The probability that the mapping selected is an injection is

Among $15$ players,$8$ are batsmen and $7$ are bowlers. Find the probability that a team of $11$ players is chosen consisting of $6$ batsmen and $5$ bowlers.

If three distinct numbers are selected from the first $100$ natural numbers,what is the probability that all selected numbers are divisible by both $2$ and $3$?

If six students,including two particular students $A$ and $B,$ stand in a row,then the probability that $A$ and $B$ are separated with one student in between them is

Vedclass Test Series

Exam Paper Generator

Online Exam Module