All words, with or without meaning, are made using all the letters of the word MONDAY. These words a

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

Think of the rank as a position in a queue. If there are $326$ people standing in front of you, what is your position? You are the $327^{th}$ person! We count all the words that come *before* $MONDAY$ lexicographically, and then add $1$ to include $MONDAY$ itself in the count.

Q: What if the word had repeating letters, like 'BANANA'? Does the method change?

Great question! If there were repeating letters, the number of permutations would not simply be $n!$. You would need to divide by the factorial of the frequency of each repeated letter (e.g., $\frac{n!}{p!q!r!}$). The logic of fixing letters remains the same, but the counting step becomes a bit more nuanced.

Q: Why do we stop at $MONA$ and not $MOND$?

We don't stop! We are systematically checking every branch of the dictionary tree. When we reach $MON$, we look at the next letter. Since $A$ comes before $D$, we count all words starting with $MONA$ first. Once we exhaust those, we move to $MOND$, which is the prefix of our target word. We are essentially walking down the dictionary path, branch by branch.

Question

All words, with or without meaning, are made using all the letters of the word $MONDAY$. These words are written as in a dictionary with serial numbers. The serial number of the word $MONDAY$ is

Options:

(A) 327

(B) 328

(C) 324

(D) 326

Sigma Percentile · Accepted Answer

### Analyzing the Alphabetical Foundation

To determine the rank of the word $MONDAY$ in a dictionary, we first identify the unique letters present in the word: $A, D, M, N, O, Y$.

Arranging these in alphabetical order, we get the sequence: $A, D, M, N, O, Y$. Since there are $6$ distinct letters, the total number of permutations is $6! = 720$.

### The 'Fix and Permute' Strategy

We systematically count all words that appear before $MONDAY$ by fixing letters in each position and calculating the permutations of the remaining letters.

For the first position, we consider letters that precede $M$ in the alphabet, which are $A$ and $D$:

*   Words starting with $A$: $5! = 120$ words.
*   Words starting with $D$: $5! = 120$ words.

Total words accounted for so far: $120 + 120 = 240$.

### Drilling Down into MONDAY

Now, we fix $M$ in the first position and move to the second slot. We look for letters that precede $O$ in the remaining set $\{A, D, N, O, Y\}$, which are $A, D,$ and $N$:

*   Words starting with $MA$: $4! = 24$ words.
*   Words starting with $MD$: $4! = 24$ words.
*   Words starting with $MN$: $4! = 24$ words.

Total words accounted for: $240 + (3 	imes 24) = 240 + 72 = 312$.

Next, we fix $O$ in the second position, giving us the prefix $MO$. We look for letters in the third position that precede $N$ from the remaining set $\{A, D, N, Y\}$, which are $A$ and $D$:

*   Words starting with $MOA$: $3! = 6$ words.
*   Words starting with $MOD$: $3! = 6$ words.

Total words accounted for: $312 + 12 = 324$.

### The Final Stretch

We fix $N$ in the third position, giving us the prefix $MON$. We look for letters in the fourth position that precede $D$ from the remaining set $\{A, D, Y\}$, which is only $A$:

*   Words starting with $MONA$: $2! = 2$ words.

Total words accounted for: $324 + 2 = 326$.

The next word in the lexicographical sequence is $MONDAY$ itself. Therefore, we add $1$ to our count to include the target word.

### The Grand Total

Summing all the counts calculated above:

$$120 + 120 + 24 + 24 + 24 + 6 + 6 + 2 + 1 = 327$$

The rank of the word $MONDAY$ is **$327$**.

Animated Solution for Mathematics - Permutations and Combinations: All words, with or without meaning, are made using all the letters of the word $M O N D A Y$ . These words are written as in a dictionary with serial numbers. The serial number of the word $M O N D A Y$ is

Select Answer:

Visualized Solution

Setting up the Dictionary

Words Starting with $A$ and $D$

Reaching the First Letter: $M$

Words with Prefix $M A$ , $M D$ , and $M N$

Reaching the Second Letter: $O$

Words with Prefix $M O A$ and $M O D$

Reaching the Third Letter: $N$

Words with Prefix $M O N A$

Reaching the Fourth Letter: $D$

Completing the Word: $M O N D A Y$

Final Rank Calculation

The Sigma Insight: Linear Permutations

Analyzing the Alphabetical Foundation

The 'Fix and Permute' Strategy

Drilling Down into MONDAY

The Final Stretch

The Grand Total

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