Acommon 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.)

This question is dealing with combinations and permutations. We need to ask: does the character depend on the value of the character before? If it does we have permutations and we add number of possible characters. If it does not we have combinations and we multiply number of possible characters. This dependance means: can we choose same character again?

In our case it does not depend and we have combinations.

One character name:
Number of combinations = 26

Two characters name:
Number of combinations = 26*36

Three characters name:
Number of combinations = 26*36*36

Four characters name:
Number of combinations = 26*36*36*36

Five characters name:
Number of combinations = 26*36*36*36*36

Six characters name:
Number of combinations = 26*36*36*36*36*36

Seven characters name:
Number of combinations = 26*36*36*36*36*36*36

Eight characters name:
Number of combinations = 26*36*36*36*36*36*36*36

To find out how many different variable names are possible we need to add these numbers:
26 + 26*36 + 26*36*36 + 26*36*36*36 + 26*36*36*36*36 + 26*36*36*36*36*36 + 26*36*36*36*36*36*36 + 26*36*36*36*36*36*36*36 =
= 26 + 936 + 33696 + 1213056 + 43670016 + 1572120576 + 56596340736 + 2037468266496 =
= 2095684645538

Then just do it (lol)