There are 2n things out of which 'n' are alike and 'n' are different, the number of ways of selecting 'n' things is 

No of like things = n

No of different things = n

No of ways of selecting any no of things from n identical things = 1 way

If we select P identical things then (n-P) different things should be selected

 No of ways of selecting n things

   = (n identical, 0 different)+ (n-1  identical, 1 different), + ….. + (0 identical, n different)

$$\begin{gathered} =1\left({\mathrm{n}}_{{\mathrm{C}}_0}\right)+1\left({\mathrm{n}}_{{\mathrm{C}}_1}\right)+ .... + 1\left({\mathrm{n}}_{{\mathrm{C}}_{\mathrm{n}}}\right) \\ =2^{\mathrm{n}} \end{gathered}$$