If the ratio of the fifth term from the beginning to the fifth term...

Given expansion is${\left(24 + \dfrac{1}{34}\right)}^{n}$

We know that general term of an expansion ${\left(x + a\right)}^{n}$ is ${T}_{r + 1} = Cr n {x}^{n - r}{a}^{r}$

(${5}^{th}$ term from beginning)

${T}_{4 + 1} = C4 n ( 24 {)}^{n - 4} \cdot {\left(\dfrac{1}{34}\right)}^{4}$

(${5}^{th}$ term from end)

We know that $r$th term from the end in a binomial expansion is ${T}_{n - r + 2}$ from the beginning.

${T}_{5}^{'} = {T}_{n - 4 + 1} = C4 n {\left(\dfrac{1}{34}\right)}^{n - 4} ( 24 {)}^{4}$

It is also given that the ratio of the fifth term from the beginning to the fifth term from the end is $\sqrt{6}: 1$.

Now, ${T}_{5}{T}_{5}^{'} = \dfrac{\sqrt{6}}{1}$

$⇒ 2\dfrac{n - 8}{4} \cdot 3\dfrac{n - 8}{4} = \sqrt{6}$

$∴ ( \sqrt{6}) \dfrac{n - 8}{2} = ( \sqrt{6} )$

$∴ n = 10$

$∴ {T}_{3} = C2 10 {\left(24\right)}^{8}{\left(\dfrac{1}{34}\right)}^{2} = \dfrac{45 \times 4}{\sqrt{3}}$

$= 60\sqrt{3}$

Hence, this is the correct option.