To find the rank of the word "DANGER", we need to consider all possible permutations of the word and find where "DANGER" falls in that list when sorted in lexicographic (dictionary) order. Here are the steps:

1. Arrange the letters in ascending order: A, D, E, G, N, R.

2. Start with the first letter of "DANGER", which is 'D'. There are two letters before 'D' in the sorted list, which are 'A' and 'C'. For each of these letters, there are 5! (factorial of 5, which is 5*4*3*2*1 = 120) ways to arrange the remaining letters. So, there are 2*120 = 240 words that start with 'A' or 'C'.

3. The next letter in "DANGER" is 'A'. There are no letters before 'A' in the remaining sorted list (D, E, G, N, R), so we don't add anything to our total.

4. The next letter is 'N'. There are two letters before 'N' in the remaining sorted list (D, E, G, R), so there are 2*4! (factorial of 4, which is 4*3*2*1 = 24) ways to arrange the remaining letters. So, there are 2*24 = 48 words that start with "DAN" followed by 'D', 'E', or 'G'.

5. The next letter is 'G'. There is one letter before 'G' in the remaining sorted list (E, R), so there are 1*3! (factorial of 3, which is 3*2*1 = 6) ways to arrange the remaining letters. So, there are 1*6 = 6 words that start with "DANG" followed by 'E'.

6. The next letter is 'E'. There are no letters before 'E' in the remaining sorted list (R), so we don't add anything to our total.

7. The last letter is 'R'. There are no letters before 'R' in the remaining sorted list, so we don't add anything to our total.

8. Add up all the possibilities: 240 (from step 2) + 48 (from step 4) + 6 (from step 5) = 294.

9. However, we have been considering all the words that come before "DANGER", so "DANGER" itself is the next word. Therefore, the rank of "DANGER" is 294 + 1 = 295.

Question

To find the rank of the word "DANGER", we need to consider all possible permutations of the word and find where "DANGER" falls in that list when sorted in lexicographic (dictionary) order. Here are the steps:

1. Arrange the letters in ascending order: A, D, E, G, N, R.

2. Start with the first letter of "DANGER", which is 'D'. There are two letters before 'D' in the sorted list, which are 'A' and 'C'. For each of these letters, there are 5! (factorial of 5, which is 5*4*3*2*1 = 120) ways to arrange the remaining letters. So, there are 2*120 = 240 words that start with 'A' or 'C'.

3. The next letter in "DANGER" is 'A'. There are no letters before 'A' in the remaining sorted list (D, E, G, N, R), so we don't add anything to our total.

4. The next letter is 'N'. There are two letters before 'N' in the remaining sorted list (D, E, G, R), so there are 2*4! (factorial of 4, which is 4*3*2*1 = 24) ways to arrange the remaining letters. So, there are 2*24 = 48 words that start with "DAN" followed by 'D', 'E', or 'G'.

5. The next letter is 'G'. There is one letter before 'G' in the remaining sorted list (E, R), so there are 1*3! (factorial of 3, which is 3*2*1 = 6) ways to arrange the remaining letters. So, there are 1*6 = 6 words that start with "DANG" followed by 'E'.

6. The next letter is 'E'. There are no letters before 'E' in the remaining sorted list (R), so we don't add anything to our total.

7. The last letter is 'R'. There are no letters before 'R' in the remaining sorted list, so we don't add anything to our total.

8. Add up all the possibilities: 240 (from step 2) + 48 (from step 4) + 6 (from step 5) = 294.

9. However, we have been considering all the words that come before "DANGER", so "DANGER" itself is the next word. Therefore, the rank of "DANGER" is 294 + 1 = 295.

Knowee AI · Accepted Answer