If all the letters of the word SCOTLAND are arranged in every possible way and…

Concept: To find the Nth arrangement of a word's letters in dictionary (alphabetical) order without listing every permutation, sort the letters alphabetically and fix them one position at a time. If k distinct letters remain to be placed, choosing each candidate letter for the next position accounts for (k − 1)! arrangements. Subtract the counts of the earlier letters block by block until the target rank falls inside the block belonging to the correct letter, and repeat this narrowing at every position.

Step-by-step for SCOTLAND (target rank = 50):

1. Sort the 8 distinct letters of SCOTLAND alphabetically: A, C, D, L, N, O, S, T.

2. Fixing A first leaves 7 letters, giving 7! = 5040 arrangements — rank 50 lies inside this block, so the first letter is A.

3. Among the remaining letters C, D, L, N, O, S, T, fixing C leaves 6 letters, giving 6! = 720 arrangements — rank 50 still lies inside, so the second letter is C.

4. Among D, L, N, O, S, T, fixing D leaves 5 letters, giving 5! = 120 arrangements — rank 50 still lies inside, so the third letter is D.

5. Among the remaining letters L, N, O, S, T, each choice of the next letter accounts for 4! = 24 arrangements: L covers ranks 1–24, N covers ranks 25–48, and O covers ranks 49–72. Rank 50 falls in the O block, so the fourth letter is O, leaving an offset of 50 − 48 = 2 within this block.

6. Among the remaining letters L, N, S, T, each choice accounts for 3! = 6 arrangements. The offset of 2 falls inside the first such block, so the fifth letter is L.

7. Among the remaining letters N, S, T, each choice accounts for 2! = 2 arrangements. The offset of 2 falls inside the first block, so the sixth letter is N.

8. Among the remaining letters S, T, each choice accounts for 1! = 1 arrangement. The offset of 2 lands on the second letter available, so the seventh letter is T, leaving S as the eighth and final letter.

Cross-check:

The two blocks ACDL and ACDN together account for the first 48 arrangements, confirming that ranks 49 and 50 sit inside the ACDO block exactly as derived above — the first word in that block is ACDOLNST (rank 49) and the next one is ACDOLNTS (rank 50).

Result: Reading off the letters chosen at each step in order — A, C, D, O, L, N, T, S — gives ACDOLNTS as the 50th word.