Unique words

[BMAD]

Wilma operates a computer with a magnetic tape drive. One day she is given a tape that contains 500,000 ”words” of four or fewer lowercase letters (consecutive words on the tape are separated by a blank character). Could all 500,000 words be different?

[DejMar]

Given that non-alphabetic letters are used except for the blank (space) to separate the "words",

Given that there are no non-alphabetic characters used, except for the blank (space) used as a word separator,

27x27x27x26 is not the correct solution; nor is 26x25x24x23 as had been selected as the best answer.

The correct count is  26x26x26x26 + 26x26x26 + 26x26 + 26.

a,b,c,d,e,f,g,...,x,y,z  = 26¹ = 26 "words" of 1 letter.
aa,ab,ac,ad,...,zx,zy,zz = 26² = 676 "words" of 2 letters.
aaa,aab,...,zzx,zzy,zzz  = 26³ = 17,576 "words" of 3 letters.
aaaa,aaab,...,zzzy,zzzz  = 26⁴ = 456,976 "words" of 4 letters.
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 456,976 + 17,576 + 676 + 26 = 475,254 different "words",

thus the magnetic tape does not contain all different "words". At least 24,746 "words" are duplicates

.

Edited March 9, 2015 by bonanova
Author's request

[bonanova]

[spoiler=Wilma needs an upgrade]Assuming they are English words having no special characters, only 358,800 words could be distinct. 26x25x24x23.

[DejMar]

There is no given restriction on the alphabet/s that is/are used, thus all 500,000 "words" could be different. 

The modern English alphabet has 26 letters, yet there are letters of other alphabets that are sometimes used in English, such as ash (Æ), oethel (Œ), and schwau (Ə). In the early modern English alphabet, the ampersand (&) - that is, and, per se, and, was once considered one of the letters of the alphabet, though it was generally used alone, but also possibly in some compound words.

Limited to the 26 letters of the modern English alphabet, absent hyphens and other symbols, there are 475,254 different words possible.  For a set of 27 letters, absent other symbols, there are 551,880 different words possible. Hence, the answer to the given problem is subject to a definition of what set of letters/characters/symbols are permitted, and what defines a "letter".

Edited March 5, 2015 by DejMar

[BMAD]

There is no given restriction on the alphabet/s that is/are used, thus all 500,000 "words" could be different. 

The modern English alphabet has 26 letters, yet there are letters of other alphabets that are sometimes used in English, such as ash (Æ), oethel (Œ), and schwau (Ə). In the early modern English alphabet, the ampersand (&) - that is, and, per se, and, was once considered one of the letters of the alphabet, though it was generally used alone, but also possibly in some compound words.

Limited to the 26 letters of the modern English alphabet, absent hyphens and other symbols, there are 475,254 different words possible.  For a set of 27 letters, absent other symbols, there are 551,880 different words possible. Hence, the answer to the given problem is subject to a definition of what set of letters/characters/symbols are permitted, and what defines a "letter".

excellent point.  I should have more clearly stated that I meant the modern English Alphabet.

[k-man]

[spoiler=Wilma needs an upgrade]Assuming they are English words having no special characters, only 358,800 words could be distinct. 26x25x24x23.

 

But, the OP says that the words can be less that 4 characters. It also doesn't say that the same character cannot be repeated inside the word.

 

Even using the alphabet with 26 unique characters there can be 500,000 distinct words. I understand that the "word" is defined as a sequence of characters of 4

or less and not a valid English word from the dictionary. So, "a", "aa", "aaa" and "aaaa" are all valid words that could appear on the tape. Using this definition of "word" there are 27x27x27x26 = 511,758 distinct words possible.

[bonanova]

Yep, brain fart. But the conclusion holds even if xxxx is permitted.

[BMAD]

[spoiler='Wilma needs an upgrade]Assuming they are English words having no special characters, only 358,800 words could be distinct. 26x25x24x23.

 

But, the OP says that the words can be less that 4 characters. It also doesn't say that the same character cannot be repeated inside the word.

 

Even using the alphabet with 26 unique characters there can be 500,000 distinct words. I understand that the "word" is defined as a sequence of characters of 4

or less and not a valid English word from the dictionary. So, "a", "aa", "aaa" and "aaaa" are all valid words that could appear on the tape. Using this definition of "word" there are 27x27x27x26 = 511,758 distinct words possible.

I disagree. 27x27x27x26 I believe implies that the first letter has 26 choices and the next three has 27 but this is not true. The second letter has 27 choices as it could be empty but the third and fourth would only have a 27th choice if the second one chose to not be the empty letter as a_a is not a word.