Transistor Series Regulator
You have an error in your equation. The calculation of \$I_e\$ is:
$$I_e=I_b+I_c = (1+\beta)\color{red}{I_b}$$
Not
$$I_e=I_b+I_c = (1+\beta)\color{red}{I_c}$$

To solve this particular problem, however, rote application of memorized equations is not the best path. Here's the thought sequence that will guide you to the solution.


The 12V zener diode on the base of the transistor probably means \$V_b\$ is being held at 12V. So check that the voltage at the cathode of the zener is greater than 12V. Looks like there's a 20V source connected to the cathode with a 220\$\Omega\$ resistor, so it's probably safe to say the zener is able to clamp at 12V (this may not end up being true if it turns out that more than 8V is dropped across the resistor).
Under normal circumstances, the emitter of a BJT will be approximately 0.7V less than the base. We already know the base is 12V, so the emitter must be 11.3V.
The current through \$R_L\$ can be found immediately using Ohm's Law:
$$I_{R_L}=\frac{V_e}{R_L}=\frac{11.3V}{1k\Omega}=11.3mA$$
Since there is no other load in the circuit, we know that all of the current through \$R_L\$ must come through the transistor. Therefore, \$I_e=11.3mA\$
Now we can use the equation you started with (with the correction I mentioned above), where \$\beta=50\$. Rearranging to solve for \$I_b\$, we get:
$$I_b = \frac{I_e}{1+\beta} = \frac{11.3mA}{1+50} = 221.6uA$$
We can do a quick check here to confirm our assumption in step #1 above. The base current of 221.6uA is the minimum current through the 220\$\Omega\$ resistor that will allow this voltage regulator to work. A quick look at Ohm's Law shows us that the theoretical minimum voltage drop across the resistor is then \$V_{min}=221.6uA*220\Omega = 49mV\$. That's far less than 8V, so we're good. Of course, the resistor will end up dropping exactly 8V because the necessary additional current will be consumed by the zener diode.
Now that we know how much current is going into the base of the transistor, we can solve how much is going through the zener diode. The only path for current to get to the base and the zener diode is through the 220\$\Omega\$ resistor. The voltages on either side of the resistor are known: 20V and 12V. Therefore, using Ohm's again, we can find how much current is going through that resistor:
$$I_R = \frac{20V-12V}{220\Omega} = 36.36mA$$

So 36.36mA is going through that resistor, of which 221.6uA is going into the base of the transistor. That leaves \$36.36mA - 221.6uA = 36.14mA\$ left for the zener diode. Looks like your solution guide apparently rounded to just 36mA.