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(C) We have the expression $\left|e^{\sum_{k=0}^n {^nC_k} \omega^k}\right|$,where $\omega$ is a cube root of unity.Using the binomial theorem,$\sum_{k

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$4$

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(C) We have the expression $\left|e^{\sum_{k=0}^n {^nC_k} \omega^k}ight|$,where $\omega$ is a cube root of unity.Using the binomial theorem,$\sum_{k=0}^n {^nC_k} \omega^k = (1+\omega)^n$.Since $1+\omega+\omega^2=0$,we have $1+\omega = -\omega^2$.Thus,the exponent is $(-\omega^2)^n = (-1)^n \omega^{2n}$.We need to evaluate $\left|e^{(-1)^n \omega^{2n}}ight|$.Let $z = (-1)^n \omega^{2n}$. Then $|e^z| = |e^{	ext{Re}(z) + i 	ext{Im}(z)}| = e^{	ext{Re}(z)}$.Case $1$: If $n$ is a multiple of $3$,say $n=3m$,then $\omega^{2n} = (\omega^3)^{2m} = 1^{2m} = 1$. So $z = (-1)^{3m} = (-1)^m$. The value is $e^{\pm 1}$.Case $2$: If $n = 3m+1$,then $\omega^{2n} = \omega^{6m+2} = \omega^2$. So $z = (-1)^{3m+1} \omega^2 = (-1)^{3m+1} (-\frac{1}{2} - i\frac{\sqrt{3}}{2})$. The real part is $\pm \frac{1}{2}$. The value is $e^{\pm 1/2}$.Case $3$: If $n = 3m+2$,then $\omega^{2n} = \omega^{6m+4} = \omega$. So $z = (-1)^{3m+2} \omega = (-1)^{3m+2} (-\frac{1}{2} + i\frac{\sqrt{3}}{2})$. The real part is $\pm \frac{1}{2}$. The value is $e^{\pm 1/2}$.Combining these,the possible values for the real part are $1, -1, 1/2, -1/2$.Thus,the possible values of the magnitude are $e^1, e^{-1}, e^{1/2}, e^{-1/2}$.There are $4$ possible values.

If $n$ is a positive integer and $\omega \neq 1$ is a cube root of unity,the number of possible values of $\left|e^{\sum_{k=0}^n {^nC_k} \omega^k}\right|$ is

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