$X$ represents sample in frequency domain and $x$ represents samples in time domain.

**NOTATION 1**

$ X[k] = \sum\limits_{n=0}^{N-1} x[n] \ e^{-j \frac{2\pi}{N} n k} $
   
$ x[n] = \frac{1}{N} \sum\limits_{k=0}^{N-1} X[k] \ e^{j \frac{2\pi}{N} n k} $



**NOTATION 2**     

$ X[k] = \frac{1}{N} \sum\limits_{n=0}^{N-1} x[n] \ e^{-j \frac{2\pi}{N} n k} $
   
$ x[n] = \sum\limits_{k=0}^{N-1} X[k] \ e^{j \frac{2\pi}{N} n k} $



**NOTATION 3**

$ X[k] = \frac{1}{\sqrt{N}} \sum\limits_{n=0}^{N-1} x[n] \ e^{-j \frac{2\pi}{N} n k} $
   
$ x[n] = \frac{1}{\sqrt{N}} \sum\limits_{k=0}^{N-1} X[k] \ e^{j \frac{2\pi}{N} n k} $


**NOTATION 4**

$ X[k] = \sum\limits_{n=0}^{N-1} x[n] \ e^{j \frac{2\pi}{N} n k} $
   
$ x[n] = \frac{1}{N}\sum\limits_{k=0}^{N-1} X[k] \ e^{-j \frac{2\pi}{N} n k} $

Please observe the scaling factor $\frac{1}{N}$ and change of a negative sign $-$ over the exponent term ${e^{\pm j\frac{{2\pi }}{N}nk} }$. Why is every notation of DFT valid?