Question

If all the words, with or without meaning, made using all the letters of the word ``RANCHI'' are arranged as in a dictionary, then the word at \(560^{\text{th}}\) position is:

• A. \text{NICHRA}
• B. \text{NICAHR}
• C. \text{NICARH}
• D. \text{NICHAR}

Hint:

For dictionary order problems, always subtract 1 from the given position and use factorial grouping to identify each letter step-by-step.

Answer 1

The Correct Option is C

The problem is about arranging the letters of the word "RANCHI" in a dictionary order and finding the word at the \(560^{\text{th}}\) position. The word "RANCHI" has 6 distinct letters: R, A, N, C, H, and I. We need to arrange these letters alphabetically first.

The alphabetical order of the letters is: A, C, H, I, N, R. 

Let's find how many total permutations can start with each letter to determine the letter that begins the word at the 560th position:

1. Starting with 'A':
   The remaining letters are C, H, I, N, R, totaling 5 letters.
   Number of permutations = \(5! = 120\).
2. Starting with 'C':
   Similarly, the number of permutations = \(5! = 120\).
3. Starting with 'H':
   Again, the number of permutations = \(5! = 120\).
4. Starting with 'I':
   Permutations = \(5! = 120\).
5. Starting with 'N':
   Again, the number of permutations = \(5! = 120\).
6. Starting with 'R':
   Permutations = \(5! = 120\).

By adding these, the positions are:

• Words starting with 'A': \(1\) to \(120\)
• Words starting with 'C': \(121\) to \(240\)
• Words starting with 'H': \(241\) to \(360\)
• Words starting with 'I': \(361\) to \(480\)
• Words starting with 'N': \(481\) to \(600\)

The \(560^{th}\) word falls under the 'N' category since it ranges from \(481\) to \(600\).

Now, find the exact position after 'N' within the subcategory:

1. Consider words starting with 'N' and then 'A': Number of permutations = \(4! = 24\). Positions: \(481\) to \(504\).
2. Consider words starting with 'N', then 'C': Number of permutations = \(4! = 24\). Positions: \(505\) to \(528\).
3. Consider words starting with 'N', then 'H': Number of permutations = \(4! = 24\). Positions: \(529\) to \(552\).
4. Consider words starting with 'N', then 'I': Number of permutations = \(= 24\) Positions: \(553\) to \(576\). The \(560^{th}\) word is within this rang\)

Let’s refine the search starting with 'NI':

1. With 'NI', next could be 'A': Number of permutations = \(3! = 6\). Positions: \(553\) to \(558\).
2. With 'NI', next 'C': Number of permutations = \(3! = 6\). Positions: \(559\) to \(564\).
   The \(560^{th}\) word is determined here.

Continue the search:

• 'NICA' gives positions: \(559\) to \(562\).

Therefore, the \(560^{th}\) position is "NICARH".

Conclusion: The word in the \(560^{\text{th}}\) position is {"NICARH"}, hence the correct answer is the option "NICARH".