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Following is my question:

If all permutations of word "FATIMAH" are written in lexicographic order. What would be the 1444th word you write?

Here is my solution:

Firstly arrange them in alphabetic order: $A,F,H,I,M,T$

Now all the permutations when first letter is A $= 6! = 720$

All the permutations when first letter is F $= dfrac6!2! = 360$ (since A is repeated twice)

All the permutations when first letter is H $= dfrac6!2! = 360$ (since A is repeated twice)

All the permutations when first letter is I $= dfrac6!2! = 360$ (since A is repeated twice)

All the permutations when first letter is M $= dfrac6!2! = 360$ (since A is repeated twice)

All the permutations when first letter is T $= dfrac6!2! = 360$ (since
A is repeated twice)

Total words up to H $= 1440$

$1441$st word = IAAFHMT

$1442$nd word = IAAFHTM

$1443$rd word = IAAFTHM

$1444$th word = IAATFHM

Please tell whether this is correct or not.

Your answer is correct until the 1443rd word.

You can think of it this way:

We know that the 1443rd word will be (obviously) $IAA****$. Now we can label $F=1, H=2, M=3, T=4$

And now the question simplifies into arranging 4-digit numbers made by $1,2,3,4$ in ascending order and taking the 4th one.

1: $1234$

2: $1243$

3: $1324$

4: $1342$

Now we can change the numbers back to the corresponding letters, and we will get $IAAFMTH$. $ _square$

There are $1440$ words with first letter $A$, $F$ or $H$, as you've correctly found out.

Using the same tactic you used to count words beginning with a single letter, count the number of words beginning with $IAAF$. You will see that there are actually six. You only wrote down three of them before moving on to $IAAT$ (which is also wrong, because $IAAH$ and $IAAM$ both come before $IAAT$ lexicographically).

Almost. Your argument is correct until you reach the $1443$rd word. Since F precedes H, H precedes M, and M precedes T in the alphabet, All the words beginning with IAAF must precede the first word beginning with IAAH, all the words beginning with IAAH must precede the first word beginning with IAAM, and all the words beginning with IAAM must precede the first word beginning with IAAT. The words beginning with IAAF are, in lexicographical order, IAAFHMT, IAAFHTM, IAAFMHT, IAAFMTH, IAAFTHM, IAAAFTMH. Thus, the $1441$st word, $1442$nd word, $1443$rd word, and $1444$th word are, respectively, IAAFHMT, IAAFHTM, IAAFMHT, IAAFMTH.