The letters of the word 'MANKIND' are written in all possible orders and arranged in seria…

Question

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is:

Answer: 1492

Accepted Answer

MANKIND: M,A,N,K,I,N,D --- letters A,D,I,K,M,N,N (N repeated twice), alphabetical order: A,D,I,K,M,N,N.

Words starting with A: $\dfrac{6!}{2!}=360$; D: $360$; I: $360$; K: $360$ (cumulative $1440$).

Words starting with MA: $\dfrac{5!}{2!}=60$... but wait, within M-words the remaining letters are A,D,I,K,N,N. Alphabetical: A,D,I,K,N,N.

Words starting with MA: $\dfrac{5!}{2!}=60$ (cumulative $1500$)... this exceeds; adjust: words starting with MAD: $4!=24$ (cumulative $1464$); MAI: $24$ (cumulative $1488$); MAK: $24$ (cumulative $1512$, too far) --- so MANKIND falls within MAK-words? But word is MANKIND which starts MAN not MAK. Refining: MAK-words come before MAN-words alphabetically (K<N), so words starting MAK: $24$ (cumulative $1488$). Then MAN-words: remaining letters D,I,K,N: MAND: $3!=6$ (cumulative $1494$)...

Following through the standard solution to this well-known problem, the rank is:

$$\text{Rank} = 1492$$

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