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A variable name can be between one and eight characters long. The first character should be one of the 26 letters, and the remaining characters can either be a letter from the 26 letters or any of the 10 digits.

The number of possible ways a situation can take place is referred to as counts of arrangements for that event. The counting rule, permutation rule, and combination rule are a few counting techniques to find the counts.

Here, repetition is allowed.

As per the length of the variable, eight different cases are possible.

Case 1: The variable has one character.

The total number of letters to choose from is equal to 26.

The number of different one character long names possible = 26.

Case 2: The variable has two characters

The number of letters to choose from for the first character = 26

The total number of possibilities for the second character is the sum of the number of letters and the number of digits =$26+10=36$

The total number of two character long names is equal to:

$26脳36=936$

The same rules apply to the remaining six cases corresponding to 3, 4, 5, 6, 7, and 8 characters long variable names.

The total number of names possible if the name is three characters long is equal to

$26脳36脳36=33696$

The total number of names possible if the name is four characters long is equal to

$26脳36脳36脳36=1213056$

The total number of names possible if the name is five characters long is equal to

$26脳36脳36脳36脳36=43670016$

The total number of names possible if the name is six characters long is equal to

$26脳36脳36脳36脳36脳36=1572120576$

The total number of names possible if the name is seven characters long is equal to

$26脳36脳36脳36脳36脳36脳36=56596340736$

The total number of names possible if the name is eight characters long is equal to

$26脳36脳36脳36脳36脳36脳36脳36=2037468266496$

Summarize the counts as follows:

The total number of different names is the sum of the different names possible for each of the one to eight characters long names.

Thus, the total number of different variable names possible is equal to 2,095,681,645,538.