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, BB, and M3477K. How many different variable names are​ possible? (Ignore the difference between uppercase and lowercase​ letters.)

Well, let's take a moment to appreciate the creativity of programmers when it comes to naming variables. Now, to answer your question, the number of possible variable names can be calculated by considering all the valid combinations for each character position.

For the first character, we have 26 options (the 26 letters of the alphabet).
For the second character, we have 36 options (26 letters + 10 digits).
For the third character, we also have 36 options.
And this goes on until the eighth character, where we still have 36 options.

To calculate the total number of possible variable names, we simply need to multiply the number of options at each position together:
26 * 36 * 36 * 36 * 36 * 36 * 36 * 36

And when we do the math, we find that there are 1,889,568,792,896 possible variable names. That's quite a lot of variables for any programmer to handle!

To find the number of different variable names possible, we need to calculate the number of options for each position in the variable name.

Let's break down the name into its components:
1. First character: Any of the 26 letters (A-Z), so there are 26 options.
2. Successive characters (up to 7 characters): Any of the 26 letters (A-Z) or any of the 10 digits (0-9), so there are a total of 26 + 10 = 36 options.

Since each position has a fixed number of options, we can simply multiply the number of options for each position to find the total number of possible variable names:
26 (options for first character) * 36 (options for the next 7 characters) = 936.

Therefore, there are 936 different variable names possible.

To find the number of different variable names that are possible, we need to calculate the total number of combinations for each character position.

For the first character, we have 26 options (26 letters of the alphabet).

For the second to eighth characters, we have 36 options (26 letters + 10 digits).

Since the length of the variable name is between 1 and 8 characters, we can calculate the total number of combinations by multiplying the number of options for each character position:

Total combinations = 26 (options for the first character) * 36^7 (options for the remaining 7 characters)

Let's calculate it:

Total combinations = 26 * (36^7)
Total combinations ≈ 26 * 78,364,164,096
Total combinations ≈ 2,033,174,872,296

Therefore, there are approximately 2,033,174,872,296 different variable names possible within the given constraints.

one letter variable ---- 26

two character variable ----- 26x36
three character variable --- 26 x 36 x 36
four character variable ---- 26 x 36 x 36 x 36 = 26(36)^3
...
eight character variable ---- 26(36)^7

Possible characters = sum of the above
= 26(1 + 36 + 36^2 + ... + 36^7)

= 26[ 1(36^8 - 1)/35 ] = appr 2.1 x 10^12 (about 2 trillion)