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 $r.$
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 $r.$
To determine the number of different possible values of $r,$ we can use the Rational Root Theorem. The Rational Root Theorem states that if a polynomial with integer coefficients has a rational root $\frac{p}{q}$ (where $p$ and $q$ are integers with no common factors), then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient.
In this case, the constant term is $24,$ and the leading coefficient is $12.$ Therefore, the possible values of $p$ are the factors of $24,$ which are $\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12,$ and $\pm24.$ The possible values of $q$ are the factors of $12,$ which are $\pm1, \pm2, \pm3, \pm4,$ and $\pm6.$
Therefore, the number of different possible values of $r$ is the product of the number of possible values of $p$ and the number of possible values of $q.$ Since there are $16$ possible values of $p$ and $10$ possible values of $q,$ the total number of different possible values of $r$ is $16\times 10 = \boxed{160}.$
To find the possible rational roots of the given polynomial, we can use the Rational Root Theorem. According to the theorem, if a polynomial with integer coefficients has a rational root $r$, it must be of the form $r = \pm \frac{p}{q}$, where $p$ is a factor of the constant term 24, and $q$ is a factor of the leading coefficient 12.
In this case, the constant term is 24, and its factors are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$. Similarly, the leading coefficient is 12, with factors $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.
So, the possible rational roots of the polynomial are:
\[r = \pm \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, \frac{4}{1}, \frac{6}{1}, \frac{8}{1}, \frac{12}{1}, \frac{24}{1},\]
\[r = \pm \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}, \frac{6}{2}, \frac{12}{2},\]
\[r= \pm \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \frac{6}{3},\]
\[r = \pm \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4},\]
\[r = \pm \frac{1}{6}, \frac{2}{6}, \frac{3}{6},\]
\[r = \pm \frac{1}{8}, \frac{2}{8},\]
\[r = \pm \frac{1}{12},\]
\[r = \pm \frac{1}{24}.\]
Simplifying these fractions, we find that the possible rational roots are:
\[r = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24.\]
Therefore, there are $\boxed{16}$ different possible values for $r$.