Find the number of words with or without meaning which can be made using all the letters of the word AGAIN. If these words are written as in a dictionary, what will be the 50th word?

Step 1: Count total permutations

First, we need to find the total number of distinct words that can be formed using all the letters of the word 'AGAIN'. The word 'AGAIN' has 5 letters in total. The letter 'A' appears 2 times, and the other letters ('G', 'I', 'N') appear once. We use the formula for permutations with repetitions, where $n$ is the total number of letters and $p_i$ is the frequency of each repeating letter.

Step 2: Calculate total number of words

Substituting the values into the formula, we have $n=5$ (total letters) and $p_1=2$ (for the letter 'A'). The factorial of 5 is $5 \times 4 \times 3 \times 2 \times 1 = 120$, and the factorial of 2 is $2 \times 1 = 2$. Dividing these gives us the total number of unique words.

Step 3: List letters in alphabetical order

To find the 50th word in dictionary order, we first list the unique letters of 'AGAIN' in alphabetical order: A, G, I, N. Note that 'A' appears twice, which is important for counting permutations.

Step 4: Count words starting with 'A'

If the first letter is 'A', we are left with the letters 'A, G, I, N'. There are 4 remaining letters, and no repetitions among them (since one 'A' is used). So, the number of words starting with 'A' is $4! = 24$.

Step 5: Count words starting with 'G'

If the first letter is 'G', we are left with the letters 'A, A, I, N'. There are 4 remaining letters, with 'A' repeating twice. So, the number of words starting with 'G' is $\frac{4!}{2!} = \frac{24}{2} = 12$.

Step 6: Count words starting with 'I' and find the 50th word

If the first letter is 'I', we are left with the letters 'A, A, G, N'. There are 4 remaining letters, with 'A' repeating twice. So, the number of words starting with 'I' is $\frac{4!}{2!} = \frac{24}{2} = 12$. Total words counted so far: $24 (A) + 12 (G) + 12 (I) = 48$. The 49th word will be the first word starting with 'N'. The letters remaining are 'A, A, G, I'. In alphabetical order, the first word starting with 'N' is 'NAAGI'. The 50th word will be the second word starting with 'N'. The letters are 'A, A, G, I'. The next word after 'NAAGI' is 'NAAIG'.

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