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Step-by-Step Solution
Step 1: Arrange the letters in alphabetical order
The letters in “MOTHER” listed alphabetically are: E, H, M, O, R, T.
Step 2: Count permutations starting with letters before M
Letters before "M" in the alphabetical list are E and H. For each of these two possible first letters, the remaining 5 letters can be arranged in 5! ways.
Total permutations so far = Number of such first letters × 5! = 2 × 5! = 2 × 120 = 240.
Step 3: Fix M as the first letter, count permutations with second letter before O
Now we fix the first letter as “M”. The remaining letters are E, H, O, R, T (in alphabetical order: E, H, O, R, T). Letters before “O” are E and H. Each choice leads to 4! permutations.
Total permutations so far = 240 + (2 × 4!) = 240 + (2 × 24) = 240 + 48 = 288.
Step 4: Fix M and O, count permutations with third letter before T
After choosing “M” and “O”, the remaining letters are E, H, R, T. In alphabetical order, these are E, H, R, T. Letters before “T” here are E, H, and R. Each choice leads to 3! permutations.
Total permutations so far = 288 + (3 × 3!) = 288 + (3 × 6) = 288 + 18 = 306.
Step 5: Fix M, O, T, count permutations with fourth letter before H
Now the remaining letters are E, H, R. Alphabetically they are E, H, R. Among these, only E is before “H”. Each such choice leads to 2! permutations.
Total permutations so far = 306 + (1 × 2!) = 306 + 2 = 308.
Step 6: Fix M, O, T, H, count permutations with fifth letter before E
The remaining letters after using M, O, T, H are E, R. Among E and R, none is before E, so 0 permutations come before fixing E.
Total permutations so far = 308 + 0 = 308.
Step 7: Add the final placement
Having fixed M, O, T, H, E, the last letter R is determined. Thus, the word “MOTHER” is the next permutation in the sequence.
Hence, the position of “MOTHER” = 308 + 1 = 309.
Therefore, the position of the word "MOTHER" in the dictionary order is 309.
Below is a reference image-based solution, preserved as provided:
