. A new programming language requires variable names to start with a letter (upper or lowercase), which can be followed by letters, numbers, or underscores. For example, a1_B_3 is a valid name. If every variable name has to be at least 3 characters but fewer than 7, how many variable names are possible?

Explanation:

Total letters available on the keyboard = 26

Doubling this amount to account for both upper and lower case letters.

Therefore, Total characters (Upper + Lower Case) = 52 characters

Total numbers on keyboard (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) = 10 characters

Underscore = 1 character

Thus, Total allowed characters = 52 + 10 + 1 = 63

This is a problem of the permutation as order of the characters results in a different name.

Our file name should be of atleast 3 and smaller than 7 (i.e. 6) characters

Therefore, total possible permutations can be obtained by subtraction permutation of 2 characters from the permutations of 6 characters.

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