Question

If all the permutations of the letters in the word ‘OBJECT’ are arranged (and numbered serially) in alphabetical order as in a dictionary, then the 717th word is:
A. TOJECB
B. TOEJBC
C. TOCJEB
D. TOJCBE

Answer 1

Hint: Combination is the number of ways in which definite objects/letters can be arranged in a defined manner. In this question, the letters of the word “OBJECT” to be arranged in the alphabetical order and the word coming at 717th position needed to be determined for which we need to use the concept of combination by evaluating the number of words formed by keeping every letter at consequent positions.

Complete step by step solution: The letters of the word “OBJECT” are arranged in alphabetical order as B C E J O T
The number of words starting with the letter “B” can be calculated $5! = 120$ .
Similarly,
The number of words starting with the letter “C” can be calculated as $5! = 120$.
Again,
The number of words starting with the letter “E” can be calculated as $5! = 120$.
Again,
The number of words starting with the letter “J” can be calculated as $5! = 120$.
Again,
The number of words starting with the letter “O” can be calculated as $5! = 120$.
Till the starting of the words with the letter “T”, a sum of $120 + 120 + 120 + 120 + 120 = 600$ words has been made.
Now, starting the formation of the word with the starting letter “T”:
The first letter after T in alphabetical order is “B”, so they will make a combination of $4! = 24$ words.
After “B”, the letter “C” will start at the second position (after “T”) to make a combination of $4! = 24$ words.
Similarly, after “C”, the letter “E” will acquire the second position (after “T”) to make a combination of $4! = 24$ words.
Again, after “E”, the letter “J” will acquire the second position (after “T”) to make a combination of $4! = 24$ words.
Again, after “J”, the letter “T” will acquire the second position (after “T”) to make a combination of $4! = 24$ words.
Till now, the total number of words formed is $600 + 24 + 24 + 24 + 24 + 24 = 720$ , but we need to determine the word at the 717th position for which we need to come back three steps as in descending order or reverse of the dictionary order.
The last word formed is TOJECB
The second last word will be TOJEBC
The third last word will be TOJCEB
And, the fourth last word (717th) will be TOJCBE.
Hence, the 717th word formed when all the permutations of the letters in the word ‘OBJECT’ are arranged (and numbered serially) in alphabetical order as in a dictionary is TOJCBE.
Option D is correct.

Note: Candidates should be aware while using the ascending order of the alphabets and while using the formula of combinations.