By the Rational Root Theorem, any rational root of the polynomial must be of the form $\frac{p}{q},$ where $p$ is a factor of 24 and $q$ is a factor of 12, in order for this root to be in simplest form. Also, $q$ must be positive.
Thus, the possibilities for $q$ are 1, 2, 3, 4, 6, and 12, and the factors for 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Thus, there are $6 \cdot 8 = \boxed{48}$ possibilities for $r.$
A polynomial with integer coefficients is of the form
\[12x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 24 = 0.\]
If $r$ is a rational root of this polynomial, then find the number of different possible values of $r.$
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