The letters of the word OUGHT are written in all possible ways and these words are arranged as in a

Q: Wait, why do we use $4!$ instead of just multiplying?

Great question! We use $4!$ (four factorial) because we are arranging four distinct letters in four distinct slots. The first slot has 4 choices, the second has 3, the third has 2, and the last has 1. Multiplying these gives $4 \times 3 \times 2 \times 1 = 24$. It is the fundamental principle of counting permutations.

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

Think of it this way: the rank of a word is its position in the list. When we sum up the counts of all words that come *before* 'TOUGH', we are counting the predecessors. Since 'TOUGH' is the next word in the sequence, we must add 1 to include the word itself in the count.

Q: What if the word had repeated letters, like 'GOOGLE'?

If there were repeated letters, the standard $n!$ formula would overcount. You would need to divide by the factorial of the frequency of each repeated letter. For example, if you had two O's, you would divide by $2!$. Always remember: distinct arrangements require division by the symmetry of repetitions!

Question

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

Options:

(A) 89

(B) 84

(C) 86

(D) 79

Sigma Percentile · Accepted Answer

### Analyzing the Alphabetical Foundation

To determine the rank of the word $TOUGH$, we first identify the set of letters involved: $\{O, U, G, H, T\}$. Sorting these alphabetically gives us the sequence: $G, H, O, T, U$.

We must count all permutations that appear lexicographically before $TOUGH$. We do this by fixing letters one by one and calculating the permutations of the remaining letters.

### Counting Words Before the T-Branch

We first consider words starting with letters that precede $T$ in the alphabet, which are $G, H,$ and $O$.

For each of these starting letters, there are $4!$ ways to arrange the remaining four letters.

$$24 + 24 + 24 = 72 	ext{ words}$$

### The T-Branch Breakthrough

Now, we fix $T$ as the first letter and examine the second position. The remaining letters are $\{G, H, O, U\}$.

Words starting with $TG$:
There are $3!$ ways to arrange the remaining three letters.
$$3! = 6 	ext{ words}$$

Words starting with $TH$:
Similarly, there are $3!$ ways to arrange the remaining three letters.
$$3! = 6 	ext{ words}$$

### The TO Refinement

We now fix $TO$ as the first two letters, matching the start of $TOUGH$. The remaining letters are $\{G, H, U\}$.

Words starting with $TOG$:
There are $2!$ ways to arrange the remaining two letters.
$$2! = 2 	ext{ words}$$

Words starting with $TOH$:
Similarly, there are $2!$ ways to arrange the remaining two letters.
$$2! = 2 	ext{ words}$$

### The Final Step

We now look for the third letter of our target, which is $U$. We fix $TOU$.

The remaining letters are $\{G, H\}$. Alphabetically, $G$ comes before $H$.

The first word formed starting with $TOU$ is $TOUGH$ itself. This accounts for the final step in our sequence.
$$1 	ext{ word}$$

### The Grand Summation

To find the final rank, we sum all the counts calculated in the previous steps:

$$72 + 6 + 6 + 2 + 2 + 1 = 89$$

The serial number (rank) of the word $TOUGH$ is **89**.

Animated Solution for Mathematics - Permutations and Combinations: The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

Select Answer:

Visualized Solution

Alphabetical Arrangement

Words starting with $G$

Words starting with $H$

Words starting with $O$

Reaching $T$ and words starting with $T G$

Words starting with $T H$

Reaching $T O$ and words starting with $T O G$

Words starting with $T O H$

The word $T O U G H$

Final Calculation

The Sigma Insight: Linear Permutations

Analyzing the Alphabetical Foundation

Counting Words Before the T-Branch

The T-Branch Breakthrough

The TO Refinement

The Final Step

The Grand Summation

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