We need the rank of the word $\text{FARMER}$ among all distinct permutations of the letters of FARMER excluding those arrangements in which the two $R$'s are adjacent.

1. Letters and alphabetical order

The word is $\text{FARMER}$ with letters: $A, E, F, M, R, R$

Alphabetical order: $A < E < F < M < R$

We must count how many valid words come before $\text{FARMER}$ in dictionary order, under the restriction that the two $R$'s are not together.

Then $\text{rank} = 1 + (\text{number of valid words before FARMER})$

2. First letter fixed comparison

We compare position by position.

The target word is: $F\;A\;R\;M\;E\;R$

So we count valid arrangements that are lexicographically smaller.

Case 1: First letter smaller than $F$

Possible smaller letters: $A, E$.

We count valid arrangements starting with each.

2.1 Starting with $A$

Remaining letters: $E, F, M, R, R$

Total distinct permutations: $\frac{5!}{2!} = 60$

Now subtract those with adjacent $RR$. Treat $RR$ as one block. Then objects are: $[RR], E, F, M$ So number of arrangements: $4! = 24$

Hence valid arrangements: $60 - 24 = 36$

2.2 Starting with $E$

Remaining letters: $A, F, M, R, R$

Similarly, $\frac{5!}{2!} - 4! = 60 - 24 = 36$

So total valid words before those starting with $F$ is $36 + 36 = 72$

3. Now first letter is $F$

We now compare the second letter with the target second letter $A$.

Current prefix: $F$ Remaining letters: $A, E, M, R, R$

We need second letter smaller than $A$. But there is no letter smaller than $A$.

So contribution here is $0$

4. Prefix $FA$

Now fix prefix $FA$. Remaining letters: $E, M, R, R$ Target third letter is $R$.

We count valid words with prefix $FA$ and third letter smaller than $R$. Possible smaller letters are: $E, M$

4.1 Prefix $FAE$

Remaining letters: $M, R, R$

Distinct permutations: $\frac{3!}{2!} = 3$ These are: $MRR, RMR, RRM$

Valid ones excluding adjacent $RR$: Only $RMR$ So count = $1$.

4.2 Prefix $FAM$

Remaining letters: $E, R, R$

Similarly, permutations are: $ERR, RER, RRE$ Only $RER$ is valid.

So count = $1$.

Hence contribution before prefix $FAR$ is $1+1=2$

So far total before $FAR\cdots$: $72+2=74$

5. Prefix $FAR$

Now remaining letters are: $E, M, R$ Target fourth letter is $M$.

We count valid arrangements with prefix $FAR$ and fourth letter smaller than $M$. Possible smaller letter: only $E$.

So consider prefix $FARE$. Remaining letters: $M, R$

Possible completions: $MR, RM$

But note: the full word already has an $R$ in third position. If we choose fourth letter $E$, then:

• completion $MR$ gives word $FAREMR$; here the two $R$'s are at positions 3 and 6, not adjacent, so valid.
• completion $RM$ gives word $FARERM$; here the two $R$'s are at positions 3 and 5, not adjacent, so valid.

Thus contribution = $2$

So total before prefix $FARM\cdots$: $74+2=76$

6. Prefix $FARM$

Remaining letters: $E, R$ Target fifth letter is $E$.

Any letter smaller than $E$? No. So contribution = $0$.

7. Prefix $FARME$

Remaining letter is $R$ only, so no further smaller choice. Contribution = $0$.

8. Final rank

Number of valid words before $\text{FARMER}$ is $76$ So the serial number is $76+1=77$

9. Comparison with stored answer

Derived answer: $77$ Stored correct answer: $77$

They match.