<p>The number of words that can be made with all the letters of the world BHARATI so that the order of consonants and the order of vowels do not change is ______.</p>
Step-by-Step Solution
Key Concept: When relative order of consonants and vowels must be preserved separately, we need to find positions for one group within a fixed arrangement, which reduces to a combination problem: choose positions for consonants (or vowels) from total positions.
<p><strong>Step 1: Identify the letters in BHARATI</strong></p><p>B, H, A, R, A, T, I → 7 letters total</p><p>Consonants: B, H, R, T (4 consonants, all distinct)</p><p>Vowels: A, A, I (3 vowels, with A repeated twice)</p><p><strong>Step 2: Understand the constraint</strong></p><p>The ORDER of consonants B-H-R-T must remain unchanged.</p><p>The ORDER of vowels A-A-I must remain unchanged.</p><p>We only choose which 4 positions (out of 7) get the consonants; vowels fill remaining positions automatically.</p><p><strong>Step 3: Apply combinations</strong></p><p>We need to select 4 positions from 7 total positions for consonants (in their fixed order).</p><p>Number of ways = C(7,4) = 7!/(4!×3!) = (7×6×5)/(3×2×1) = 35</p><p>Once consonant positions are fixed, vowels A, A, I arrange themselves in the remaining 3 positions in only 1 way (since their relative order is fixed as A-A-I).</p><p><strong>Step 4: Final answer</strong></p><p>∴ Answer: <strong>35</strong></p>
Correct Answer: 35