I am solving a problem: How many words are less than four letters long and contain only the letters A, B, C, D, and E? Here, ‘word’ refers to any string of letters.

**My Solution 1 uses Generating functions** gives the wrong answer

(1+x+x^2+x^3) * (1+x+x^2+x^3) * (1+x+x^2+x^3) * (1+x+x^2+x^3) * (1+x+x^2+x^3)

where (1+x+x^2+x^3) is Generating function for each letter. This approach gave a wrong answer (35+15+5+1) which is the sum of coffecients of (x^3, x^2, x, 1) in x^15 + 5 x^14 + 15 x^13 + 35 x^12 + 65 x^11 + 101 x^10 + 135 x^9 + 155 x^8 + 155 x^7 + 135 x^6 + 101 x^5 + 65 x^4 + 35 x^3 + 15 x^2 + 5 x + 1

**Correct Approach using Case Work** gives the correct answer

Case 1: The word is one letter long. Clearly, there are 5 of these words.

Case 2: The word is two letters long. Constructing the set of these words, there are 5 options for the first letter and 5 options for the second letter, so there are 5^2 = 25 of these words.

Case 3: The word is three letters long. By similar logic as above, we have 5 options for the first letter, 5 options for the second, and 5 options for the third. Then there are 5^3 = 125 of these letters.

Adding all our cases up, there are 5 + 25 + 125 = 155 words that are less than four letters long and contain only the letters A, B, C, D, and E.

**Could Someone help me in explaining why the Generating function approach failed here?**