Question

Computer Variable Names A common computer programming rule is that names of variables must be between one and eight characters long. The first character can be any of the 26 letters, while successive characters can be any of the 26 letters or any of the 10 digits. For example, allowable variable names include A, BBB, and M3477K. How many different variable names are possible? (Ignore the difference between uppercase and lowercase letters)

Short answer

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

Step-by-step solution

Given information

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.

State the counting rule

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.

Compute the counts for each length

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:

$26\times 36=936$

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

$26\times 36\times 36=33696$

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

$26\times 36\times 36\times 36=1213056$

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

$26\times 36\times 36\times 36\times 36=43670016$

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

$26\times 36\times 36\times 36\times 36\times 36=1572120576$

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

$26\times 36\times 36\times 36\times 36\times 36\times 36=56596340736$

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

$26\times 36\times 36\times 36\times 36\times 36\times 36\times 36=2037468266496$

Summarize the counts as follows:

Length of the variable	Number of ways to name
1	26
2	936
3	33,696
4	1,213,056
5	43,670,016
6	1,572,120,576
7	56,596,340,736
8	2,037,468,266,496

Compute the total counts of names

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

Length of the variable	Number of ways to name
1	26
2	936
3	33,696
4	1,213,056
5	43,670,016
6	1,572,120,576
7	56,596,340,736
8	2,037,468,266,496
Total	2,095,681,645,538

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