Every well order of the real numbers has some order type between $\mathfrak{c}$ and $\mathfrak{c}^+$, and for any given order type arising, every permutation of $\mathbb{R}$ induces another well order of the same order type. So the total number of well-orders of $\mathbb{R}$ is exactly $$\mathfrak{c}^+\cdot\mathfrak{c}^{\mathfrak{c}}=\mathfrak{c}^{\mathfrak{c}}=2^{\mathfrak{c}}=\beth_2.$$

Similarly, for any infinite cardinal $\kappa$, there will be $\kappa^+$ many possible order types, and again for each order type we have $\kappa^\kappa$ many rearrangements of it, making $\kappa^+\cdot\kappa^\kappa=\kappa^\kappa=2^\kappa$ many well orders in all.

Every well order of the real numbers has some order type between $\mathfrak{c}$ and $\mathfrak{c}^+$, and for any given order type arising, every permutation of $\mathbb{R}$ induces another well order of the same order type and conversely. So the total number of well-orders of $\mathbb{R}$ is exactly $$\mathfrak{c}^+\cdot\mathfrak{c}^{\mathfrak{c}}=\mathfrak{c}^{\mathfrak{c}}=2^{\mathfrak{c}}=\beth_2.$$

Similarly, for any infinite cardinal $\kappa$, there will be $\kappa^+$ many possible order types, and again for each order type we have $\kappa^\kappa$ many rearrangements of it, making $\kappa^+\cdot\kappa^\kappa=\kappa^\kappa=2^\kappa$ many well orders in all.