Determining the Dictionary Rank of a Specific Arrangement of Letters
<p>Let's calculate the rank of the word RACHIT.</p>
<p>Step 1: Write down the letters in alphabetical order: A, C, H, I, R, T.</p>
<p>Step 2: Fix the first letter and calculate the number of permutations for the remaining letters.</p>
<p>We have 5 positions left, so we will calculate \(5!\) permutations for the following scenarios:</p>
<ul>
<li>R is fixed (A, C, H, I, T): count the permutations starting with A, C, H, and I.</li>
</ul>
<p>\(4 \times 5! = 4 \times 120 = 480\)</p>
<p>Step 3: Now fix the second letter and calculate permutations for the remaining letters, until the selected letter exceeds the corresponding letter in RACHIT.</p>
<p>For the word RACHIT, after R, A is fixed:</p>
<ul>
<li>R, A is fixed (C, H, I, T): count the permutations starting with C and H.</li>
</ul>
<p>\(2 \times 4! = 2 \times 24 = 48\)</p>
<p>Step 4: Repeat step 3 for all subsequent letters.</p>
<ul>
<li>R, A, C is fixed (H, I, T): count the permutations starting with H.</li>
</ul>
<p>\(1 \times 3! = 1 \times 6 = 6\)</p>
<p>Finally, add all these permutations:</p>
<p>480 (for R) + 48 (for RA) + 6 (for RAC) + 1 (for the next turn RACHI to RACHIT)</p>
<p>Total rank = 480 + 48 + 6 + 1 = 535</p>
<p>So the rank of the word RACHIT is 535.</p>
