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Textbook Explanation:

Strings are made up from a prescribed alphabet symbols. If the prescribed alphabet consists of the symbols 0, 1, and 2 ... In general, if n is any positive integer, then by the rule of product there are $3^n$ strings of length n for the alphabet 0, 1, and 2.

Logical Question:

  1. Does 3 inside $3^n$ refer to the three alphabets 0, 1, 2 ?

  2. If the answer to last question is yes, does it mean if there are alphabets of 0, 1, 2, and 3. Then there are $4^n$ strings of length n ?

  3. What would be the general formula ? Would it be $infinity^n$?
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    Yes and yes $~~~$ – 2017-02-18
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    And also the answer to the 3rd question I just updated is yes too ? – 2017-02-18
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    "general formula" for what? For the number of strings possible with an infinite number of choices for each alphabet symbol? – 2017-02-18
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    Yes, like if 0 , 1, 2, continues until infinity and the length of the string is unknown, so considering n. Then we have $infinity^n$ ? – 2017-02-18
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    I think that question is rather pointless... $\infty^n$ is as much of an infinity as $\infty$ is... and there are an infinite number of length $1$ strings using an alphabet with infinitely many characters.... the more important thing to note is that for an alphabet with $k$ characters there are $k^n$ strings of length $n$... we ever only really consider alphabets with a finite number of letters as those are the only ones that are generally interesting. – 2017-02-18

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