The letters of word LABOUR are permuted in all possible ways and the GMAT Problem Solving

Question: The letters of word LABOUR are permuted in all possible ways and the words thus formed are arranged in alphabetical order, then what is the rank of the word LABOUR?

A. 132
B. 152
C. 242
D. 340
E. 400

Answer: C

Approach Solution (1):
The order of each letter in the dictionary is ABLORU.
Now, with A in the beginning, the remaining letters can be permuted in 5! ways.
Similarly, with B in the beginning, the remaining letters can be permuted in 5! ways.
With L in the beginning, the first word will be LABORU, the second will be LABOUR.
Hence, the rank of the word LABOUR is,
5! + 5! + 2 = 120 + 120 + 2 = 242
Correct option: C

Approach Solution (2):
The letters of word LABOUR arranged in alphabetical order are A B L O R U
To rank the words in alphabetical order we need to fix the first letter and try rearranging remaining letters in alphabetical order
When the first letter is A, the remaining letters can be arranged in 5! = 120 ways, thus resulting in total 120 words that start with A
Then we would have B as the first letter and remaining letters can be arranged in 5! = 120 ways
Thereby resulting in a total of 120 words that start with B
Next we have words starting with L
The first word formed in alphabetical order, starting with L, is = L A B O R U
Subsequent words in the rank can be obtained by changing the position of alphabets from the rightmost position.
The next word in the rank is LABOUR
Hence, the rank of word LABOUR is 120 + 120 + 2 = 242
Correct option: C

Approach Solution (3):
Alphabets present in the word LABOUR are (L, A, B, O, U, R)
Order of these letters according to the dictionary is A, B, L, O, R, U
With A in the beginning (A _ _ _), the remaining letters can be permuted in 5! Ways
Similarly, with B in the beginning (B _ _ _), the remaining letters can be permuted in 5! Ways
With L in the beginning (L _ _ _), the first word will be LABORU
The second will be LABOUR
Hence the rank of the word LABOUR is 5! + 5! + 2 = 242
Correct option: C

“The letters of word LABOUR are permuted in all possible ways and the words thus formed are arranged in alphabetical order, then what is the rank of the word LABOUR?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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