The average of the two digit numbers, which remain the same when the digits interchange their positions, is:
$\left( A \right) 33$
$\left( B \right) 44$
$\left( C \right) 55$
$\left( D \right) {\text{ }}66$

Hint: Before we go to the solution first of all we know about the average concept.
 Average is the sum of the data values divided by the total number of data values.
First we find out the two digit numbers which remain the same when the digits interchange their position and then we find the average of this number.
 Finally we get the required answer.

Formula used: Average = $\dfrac{{{\text{sum of these data values}}}}{{{\text{number of data values}}}}$

Complete step-by-step solution:
It is given that the two digit numbers which remain same when the digits interchange their positions are
${\text{11, 22, 33, 44, 55, 66, 77, 88, 99}}$
Here the total number of terms of these two digit numbers is ${\text{9}}$.
Now we have to sum of these two digit numbers = ${\text{11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99}}$
On adding we get,
\[ \Rightarrow {\text{ 495}}\]
Now by using the formula and we get
Average = $\dfrac{{{\text{sum of these data values}}}}{{{\text{number of data values}}}}$
$ \Rightarrow \dfrac{{{\text{495}}}}{{\text{5}}}$
On dividing we get,
\[ \Rightarrow {\text{ 55}}\]
Therefore the average of the two digit numbers, which remain the same when the digits interchange their positions, is $55$.

Hence the correct option is $\left( {\text{C}} \right)$

Note: Average can be defined as the central value in a set of data.
In other words, an average value represents the middle value of a data set.
Here the data sets are two digit numbers which remain same when the digits interchange their position.
That is ${\text{11, 22, 33, 44, 55, 66, 77, 88, 99}}$.
Here note that the middle value is ${\text{55}}$.
This is the correct answer.
This method helps to solve the question quickly and efficiently which can save a lot of time.