The words with or without meaning 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?

Question

Accepted Answer

To determine the 50th word when all permutations of the letters in "AGAIN" are arranged in dictionary order, we need to systematically count the permutations starting from the beginning.

1. **Understanding the Word "AGAIN":**
   - The word "AGAIN" consists of 5 letters: A, G, A, I, N.
   - The letters include a repetition of 'A', which affects the total number of unique permutations.

2. **Calculating Total Permutations:**
   - The total number of permutations of the letters in "AGAIN" is calculated using the formula for permutations of multiset: 
     $$
     \frac{5!}{2!} = \frac{120}{2} = 60
     $$
   - This means there are 60 unique permutations.

3. **Arranging in Dictionary Order:**
   - Dictionary order means arranging the permutations as if they were in a dictionary, starting with the smallest lexicographical order.

4. **Counting Permutations:**
   - Start with the smallest letter 'A'. The permutations starting with 'A' are:
     - AAGIN, AAGNI, AAI, AAN, AG, AI, AN (and so on).
   - The number of permutations starting with 'A' is:
     $$
     \frac{4!}{1!} = 24
     $$
   - Next, consider permutations starting with 'G':
     - GAAIN, GAA, GAI, GAN, GI, GN (and so on).
   - The number of permutations starting with 'G' is:
     $$
     \frac{4!}{2!} = 12
     $$
   - Next, consider permutations starting with 'I':
     - IAA, IAG, IAN, IG, IN (and so on).
   - The number of permutations starting with 'I' is:
     $$
     \frac{4!}{2!} = 12
     $$
   - Next, consider permutations starting with 'N':
     - NAAIG, NAAGI, NAI, NAG, NI (and so on).
   - The number of permutations starting with 'N' is:
     $$
     \frac{4!}{2!} = 12
     $$

5. **Finding the 50th Word:**
   - Adding up the permutations:
     - 24 (starting with 'A') + 12 (starting with 'G') + 12 (starting with 'I') = 48
   - The 49th permutation starts with 'N', and the 50th permutation is the second permutation starting with 'N'.

6. **Conclusion:**
   - The 50th word is the second permutation starting with 'N', which is "NAAIG".

**Explanation of Incorrect Options:**
- **Option 2 (NAAGI):** This is the first permutation starting with 'N', making it the 49th word.
- **Option 3 (NIAAG):** This permutation would come after permutations starting with 'NAA', making it later than the 50th position.
- **Option 4 (NGAAI):** This permutation would come after permutations starting with 'NAA' and 'NAI', making it later than the 50th position.

Thus, Option 1, "NAAIG," is correctly identified as the 50th word in dictionary order.