Bjt base current calculation
First try to read this 


  The transistor going into saturation isn't a property of the
  transistor itself, but instead a property of the circuit surrounding
  the transistor and the transistor, as part of it.


A question about Vce of an NPN BJT in saturation region



For this circuit with ideal transistor (current controlled current source CCCS) any base current large than:

\$\Large I_B &gt; \frac{\frac{V_{cc}}{R_c}}{\beta }\$ will saturate the BJT.

But in real life, ideal transistor don't exist. For any real world transistor, the β is not constant. β varies with Ic, Vce, temperature. And what is worse, every single transistor will have different beta value and beta will changes for different operating conditions also.
Also in saturation \$I_C = I_B * β\$ do not hold anymore.

So to overcome this problem with beta and saturation we are forced to use "overdrive factor" or "Forced Beta" trick.

We simply increase the base current well beyond \$I_B &gt; \frac{\frac{V_{cc}}{R_C}}{\beta }\$. 

We do this to make sure that we have enough base current to put the transistor well into saturation for every condition we have in our circuit.

Additional most BJT's vendors define saturation region when Ic/Ib = 10 (called Forced Beta). And the most data-sheet show Vce_sat for Ic/Ib = 10

So, to be one hundred percent sure that your BJT will be in saturation region you must use this so-called forced beta technique when choosing base resistor value.

$$\frac{I_C}{I_B} = 10$$

$$R_B = \frac{V_{IN} - V_{BE}}{0.1*I_C}$$

$$R_C = \frac{V_{CC} - V_{CE_{sat}}}{I_C}$$

Or we can use KVL and solve for \$R_B\$ 

$$I_B=\frac{V_{IN} - V_{BE}}{R_B}$$

$$V_{CE} = V_{CC} - I_C*R_C = V_{CC} - \beta*I_B*R_C = V_{CC} - \beta \frac{V_{IN} - V_{BE}}{R_B} * R_C $$ Solving for \$R_B\$

$$R_B\leqslant \frac{V_{IN_{min}} - V_{BE}}{V_{CC} - V_{CE_{sat}}}*\frac{\beta_{min}}{K}*R_C$$

And K = 3...10 - overdrive factor