Problem 84

To determine the number of irrational numbers among the given set, we analyze each number individually:

1. The number $2$ is an integer, and all integers are rational numbers because they can be expressed as a fraction $\frac{a}{1}$, where $a$ is an integer. Therefore, $2$ is rational.

2. The number $0$ is also an integer, and by the same reasoning as above, it is rational.

3. The number $\sqrt{5}$ cannot be expressed as a fraction of two integers, as there are no two integers whose ratio squared equals $5$. Therefore, $\sqrt{5}$ is irrational.

4. The number $\frac{\pi}{3}$ involves $\pi$, which is a well-known irrational number. Since the ratio of an irrational number to a rational number (in this case, $3$) remains irrational, $\frac{\pi}{3}$ is irrational.

5. The number $\sqrt[3]{27} = 3$ is the cube root of $27$, which equals $3$. Since $3$ is an integer, it is rational.

6. The number $0.1010010001\ldots$ (with one more $0$ between every two $1$'s) represents an infinite, non-repeating decimal. This pattern does not repeat in a regular manner that would allow it to be expressed as a fraction of two integers. Therefore, it is irrational.

Summarizing the findings:

- Rational numbers: $2$, $0$, $\sqrt[3]{27} = 3$
- Irrational numbers: $\sqrt{5}$, $\frac{\pi}{3}$, $0.1010010001\ldots$

Thus, the total number of irrational numbers in the given set is $3$.

Therefore, the correct answer is $\boxed{C}$.