Answer: C

Video solution:

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Written solution:

The reciprocals of $1$, $2$, and $4$ are $\dfrac11$, $\dfrac12$, and $\dfrac14$, respectively. The average of these reciprocals is $$\begin{align*}\dfrac{\left(1 + \dfrac12 + \dfrac14\right)}{3} &= \dfrac{\left(\dfrac{7}{4}\right)}{3} \\&= \dfrac{7}{12}. \end{align*}$$

As the harmonic mean is the reciprocal of the average of the reciprocals of the numbers (which we just calculated to be $\dfrac{7}{12}$), we conclude that the harmonic mean is $\dfrac{12}{7}.$

Thus, the correct answer is C.