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The name of a variable in the JAVA programming language is a string of $1$ to $65,535$ characters, where each character can be an uppercase or a lowercase letter, a dollar sign, an underscore, or a digit. Furthermore, the first character in the string must not be a digit. How many different variable names are there in JAVA?

Hint Given:

I know there are the following characters:

  • $26$ Lowercase Letters
  • $26$ Uppercase Letters
  • $1$ Dollar Sign ($) - $1$ Underscore(_)

For a total of $54$ ways.

I am unsure where to go from this point. If this could be worked out with an explanation it would be much appreciated.

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    $\begingroup$ Have you tried working out the count of variable names for a couple characters, say, $1,2,3,4,\dots$ characters in length? $\endgroup$ – abiessu Apr 25 '18 at 2:25
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Hint: Do you recognize the exponent on $r$ as the number of characters after the first? How many one character names are there? What does that represent?

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  • $\begingroup$ That makes sense. If the string has only one character, there are 54 possibilities since the first character must not be a digit. If the string has two characters, there are 54 * 64 possibilities and so on. So would the geometric series be 54 * 1 * (64^65535 - 1) / (64 - 1) equaling 54 * (64^65535 - 1) / 63? $\endgroup$ – John Larkos Apr 25 '18 at 2:57
  • $\begingroup$ That is correct. $\endgroup$ – Ross Millikan Apr 25 '18 at 3:03

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