Given the mathematical problem, we need to find the least positive integer $n$ for which there exists a set of distinct positive integers $\{s_1, s_2, \ldots, s_n\}$ such that:

$$\left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.$$

First, observe that the expression $\left( 1 - \frac{1}{s_i} \right) = \frac{s_i - 1}{s_i}$. Therefore, the problem can be rewritten as:

$$\frac{(s_1 - 1)(s_2 - 1) \cdots (s_n - 1)}{s_1 s_2 \cdots s_n} = \frac{51}{2010}.$$

This equation can be rearranged as:

$$(s_1 - 1)(s_2 - 1) \cdots (s_n - 1) = \frac{51}{2010} \times s_1 s_2 \cdots s_n.$$

Simplifying the fraction $\frac{51}{2010}$:

- The greatest common divisor of 51 and 2010 is 3.

We divide both the numerator and denominator by 3:

$$\frac{51}{2010} = \frac{17}{670}.$$

Thus, our equation becomes:

$$(s_1 - 1)(s_2 - 1) \cdots (s_n - 1) = \frac{17}{670} \times s_1 s_2 \cdots s_n.$$

This implies:

$$670(s_1 - 1)(s_2 - 1) \cdots (s_n - 1) = 17 s_1 s_2 \cdots s_n.$$

Therefore, we have:

$$670 \prod_{i=1}^{n} (s_i - 1) = 17 \prod_{i=1}^{n} s_i.$$

The left-hand side and the right-hand side must equal in factor counts, compensating for the prime factors. The smallest $n$ would be determined by choosing the minimal possible distinct values for $s_1, s_2, \ldots, s_n$.

After trial by substitution of small integers and ensuring integer solutions exist for all conditions, you find that $n = 39$ satisfies the equation as the least number of set members to solve:

$$\boxed{39}.$$