How many license plates can be made using either two or three letters followed by either two or three digits?

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Number of license plates having either two or three letters followed by either two or three numbers:

Case 1: 2 letters followed by 2 numbers are $26^{2} \times 10^{2}$

Case 2: 2 letters followed by 3 numbers are $26^{2} \times 10^{3}$

Case 3: 3 letters followed by 2 numbers are $26^{3} \times 10^{2}$

Case 4: 3 letters followed by 3 numbers are $26^{3} \times 10^{3}$

So, the total number of license plates having two or three letters followed by two or three numbers

$=\left(26^{2} \times 10^{2}\right)+\left(26^{2} \times 10^{3}\right)+\left(26^{3} \times 10^{2}\right)+\left(26^{3} \times 10^{3}\right)=20,077,200$