Let the three consecutive integers be represented as $x-1$, $x$, and $x+1$.

According to the problem, the sum of these integers is equal to their product: $(x-1) + x + (x+1) = (x-1) \times x \times (x+1)$

Simplifying the left-hand side: $(x - 1) + x + (x + 1) = 3x$

Simplifying the right-hand side: $(x - 1) \times x \times (x + 1) = x(x^2 - 1) = x^3 - x$

Now, equating both sides: $3x = x^3 - x$

Rearranging the equation: $x^3 - 4x = 0$

Factoring out x: $x(x^2 - 4) = 0$

This gives two possible factors: $x = 0$ or $x^2 - 4 = 0$

Solving $x^2 - 4 = 0$: $x^2 = 4 \rightarrow x = 2$ or $x = -2$

Thus, the possible values for x are 0, 2, and -2.

Conclusion: There are 3 such possibilities: $x = 0$, $x = 2$, and $x = -2$.

Hence, the correct answer is C. Only three.