1. Arrange the letters in dictionary order

The word is OUGHT, whose letters are all distinct:

$\{O, U, G, H, T\}$

In alphabetical order:

$G < H < O < T < U$

We need the rank (serial number) of the word TOUGH among all permutations arranged lexicographically.

2. Count words before TOUGH

We count how many valid permutations come before TOUGH.

Position 1: T

Letters smaller than $T$ available are:

$G, H, O$

For each such choice at the first position, the remaining 4 letters can be arranged in:

$4! = 24$

So contribution is:

$3 \times 24 = 72$

Position 2: O

Now first letter is fixed as $T$. Remaining letters are:

$\{G,H,O,U\}$

We want second letter $O$. Letters smaller than $O$ among remaining are:

$G, H$

For each such choice, remaining 3 letters can be arranged in:

$3! = 6$

So contribution is:

$2 \times 6 = 12$

Total so far:

$72 + 12 = 84$

Position 3: U

Now first two letters fixed as $TO$. Remaining letters are:

$\{G,H,U\}$

We want third letter $U$. Letters smaller than $U$ are:

$G, H$

For each such choice, remaining 2 letters can be arranged in:

$2! = 2$

So contribution is:

$2 \times 2 = 4$

Total so far:

$84 + 4 = 88$

Position 4: G

Now first three letters fixed as $TOU$. Remaining letters are:

$\{G,H\}$

We want fourth letter $G$. Letters smaller than $G$ among remaining: none.

Contribution:

$0$

Position 5: H

Only one letter remains, so contribution is:

$0$

3. Find the rank

Rank is one more than the number of words before it:

$88 + 1 = 89$

So the serial number of TOUGH is:

$\boxed{89}$

4. Option check

• A: $79$
• B: $84$
• C: $89$ ✅
• D: $86$

Hence the correct option is C.

5. Comparison with stored answer

Stored correct answer: C

Derived answer: C

They agree.