The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as

Q: Why do we divide by $2!$ in the calculation $\frac{6!}{2!}$?

Great question! In permutations, if we have identical items, swapping them doesn't create a new unique word. Since we have two $N$s, the total arrangements of $6$ letters would be $6!$ if all were distinct. But because the two $N$s are indistinguishable, we divide by $2!$ to remove the redundant counts. It's like counting identical twins as one person—you have to adjust the total!

Q: Why do we add $1$ at the very end of the calculation?

Think of the rank as the position in a list. If you are the $1491$st person in line, and you join, you are the $1492$nd. The rank is the number of words *before* our target word plus the word itself. We count all the permutations that precede 'MANKIND' and then add $1$ to include 'MANKIND' in the count.

Q: What if the word had three $N$s instead of two?

The logic remains identical! If you had three $N$s, you would divide by $3!$ instead of $2!$ to account for the indistinguishable arrangements. The 'Fix and Permute' strategy is universal—it doesn't matter how many repetitions you have, just adjust your denominator to match the factorial of the count of each repeated letter.

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 ______.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To find the rank of the word $MANKIND$, we must treat the dictionary as a machine of strict alphabetical order. First, we identify the letters in the word: $M, A, N, K, I, N, D$.

There are $7$ letters in total. Sorting these alphabetically, we obtain the pool: $A, D, I, K, M, N, N$.

Note that the letter $N$ repeats twice. We must account for this repetition by dividing by $2!$ when calculating permutations of the remaining letters.

### The 'Fix and Permute' Strategy

We determine how many words precede $MANKIND$ by fixing letters one by one. The target word starts with $M$.

The dictionary lists all words starting with $A, D, I,$ and $K$ before reaching $M$. For each of these $4$ starting letters, we have $6$ remaining positions to fill with the remaining letters (including two $N$s).

The number of arrangements for each is:

$$
\frac{6!}{2!} = \frac{720}{2} = 360
$$

Since there are $4$ such letters, the total number of words preceding those starting with $M$ is:

$$
4 	imes 360 = 1440
$$

### The Deep Dive: Matching and Freezing

We now fix $M$ and move to the second letter, $A$. Since $A$ is the first available letter in our sorted pool, it matches our target. We freeze $MA$.

Next, we look for the third letter, $N$. Before $N$, the dictionary lists words starting with $MAD, MAI,$ and $MAK$. For each of these $3$ prefixes, we have $4$ letters remaining (including two $N$s).

The number of arrangements for each is:

$$
\frac{4!}{2!} = \frac{24}{2} = 12
$$

Thus, we bypass $3 	imes 12 = 36$ words.

### The Final Stretch

We match $N$ and freeze $MAN$. The remaining pool is $\{D, I, K, N\}$. We need $K$ as the fourth letter.

Before $K$, we have $D$ and $I$. Words starting with $MAND$ and $MANI$ must be bypassed. For each, there are $3! = 6$ arrangements:

$$
6 + 6 = 12 	ext{ words}
$$

Next, we match $K$ and freeze $MANK$. The remaining pool is $\{D, I, N\}$. We need $I$ as the fifth letter. Before $I$, we have $D$. Words starting with $MANKD$ must be bypassed:

$$
2! = 2 	ext{ words}
$$

Finally, we match $I$ and freeze $MANKI$. The remaining pool is $\{D, N\}$. We need $N$ as the sixth letter. Before $N$, we have $D$. Words starting with $MANKID$ must be bypassed:

$$
1! = 1 	ext{ word}
$$

### The Grand Summation

We have now reached the word $MANKIND$ itself, which counts as the final $1$ position. Summing all the bypassed words and the word itself:

$$
1440 + 36 + 12 + 2 + 1 + 1 = 1492
$$

The rank of the word $MANKIND$ is **1492**.

Animated Solution for Mathematics - Permutations and Combinations: 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 ______.

Enter Numerical Value:

Visualized Solution

Analyzing the Word

Alphabetical Sorting

Words Starting with $A$

Words Starting with $D, I, K$

Fixing the First Letter: $M$

Fixing the Second Letter: $A$

Third Letter: Bypassing $D, I, K$

Fixing the Third Letter: $N$

Fourth Letter: Bypassing $D, I$

Fixing the Fourth Letter: $K$

Fifth Letter: Bypassing $D$

Fixing the Fifth Letter: $I$

Sixth Letter: Bypassing $D$

The Final Word: MANKIND

Calculating the Total Rank

The Sigma Insight: Linear Permutations

Analyzing the Setup

The 'Fix and Permute' Strategy

The Deep Dive

The Final Stretch

The Grand Summation

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