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Step-by-Step Solution
Step 1: Arrange the Letters Alphabetically
The word is "MANKIND". First, list the letters in alphabetical order: A, D, I, K, M, N, N.
Step 2: Assign Numbers to Each Alphabet
Assign each letter a position number, starting from 0:
A → 0, D → 1, I → 2, K → 3, M → 4, N → 5, N → 6.
Step 3: Process the First Letter "M"
In "MANKIND", the first letter is "M". From our alphabet list, "M" is at position 4. The remaining 6 letters after "M" in the word are "ANKIND", which can be arranged in
\frac{6!}{2!} ways (because there are 2 "N"s).
Thus, this contributes
4 \times \frac{6!}{2!}
to the rank.
Step 4: Remove "M" and Update Alphabet Box
After counting the arrangements contributed by "M", remove "M" from the list. We create a new alphabet box without "M". Now the positions will be:
A → 0, D → 1, I → 2, K → 3, N → 4, N → 5.
Step 5: Process the Second Letter "A"
In the new box, "A" is at position 0. The remaining 5 letters are "NKIND", which can be arranged in
\frac{5!}{2!} ways (because there are still 2 "N"s in total).
Hence, the contribution from "A" is
0 \times \frac{5!}{2!} = 0.
Step 6: Remove "A" and Update Alphabet Box
Delete "A". The updated list is now: D → 0, I → 1, K → 2, N → 3, N → 4.
Step 7: Process the Third Letter "N"
In this box, "N" appears at positions 3 and 4. We always choose the first occurrence, so we take position 3. The remaining letters "KIND" can be arranged in
\frac{4!}{2!} ways. (Now we have 1 "N" in "KIND", plus the one we're using, so effectively 2 identical N's overall.)
Hence contribution is
3 \times \frac{4!}{2!}.
Step 8: Remove This "N" and Update Alphabet Box
Removing the "N" at position 3 leaves: D → 0, I → 1, K → 2, N → 4.
Step 9: Process the Fourth Letter "K"
"K" is at position 2 in this new box. The remaining letters "IND" can be arranged in
3! ways.
Hence contribution is
2 \times 3!.
Step 10: Remove "K" and Update Alphabet Box
After removing "K": D → 0, I → 1, N → 2.
Step 11: Process the Fifth Letter "I"
"I" is at position 1 in the new list. The remaining letters "ND" can be arranged in
2! ways.
So the contribution is
1 \times 2!.
Step 12: Remove "I" and Update Alphabet Box
Deleting "I" leaves: D → 0, N → 1.
Step 13: Process the Sixth Letter "N"
"N" is at position 1 in this box. The remaining single letter "D" can be arranged in
1! ways.
Hence contribution is
1 \times 1!.
Step 14: Remove This "N" and Check Remaining Letters
Removing "N" leaves only "D" at position 0.
Step 15: Process the Last Letter "D"
"D" is at position 0 in the new list, and there are no letters left to arrange. By convention, the final letter contributes
0! = 1.
Combine All Contributions
The rank of "MANKIND" is:
4 \times \frac{6!}{2!} \;+\; 0 \times \frac{5!}{2!} \;+\; 3 \times \frac{4!}{2!} \;+\; 2 \times 3! \;+\; 1 \times 2! \;+\; 1 \times 1! \;+\; 0!
\frac{6!}{2!} = \frac{720}{2} = 360,\quad \frac{5!}{2!} = \frac{120}{2} = 60,\quad \frac{4!}{2!} = \frac{24}{2} = 12,\quad 3! = 6,\quad 2! = 2,\quad 1! = 1,\quad 0! = 1.
Hence:
4 \times 360 + 0 \times 60 + 3 \times 12 + 2 \times 6 + 1 \times 2 + 1 \times 1 + 1 = 1440 + 0 + 36 + 12 + 2 + 1 + 1 = 1492.
Final Answer
The serial number (dictionary rank) of the word "MANKIND" is 1492.
