<p>The letters of the word 'MUMBAI' are arranged in all possible ways as in a dictionary, the rank of the word 'MUMBAI' is</p>
Step-by-Step Solution
Key Concept: When letters repeat, use the formula for permutations with repetition: n!/n₁!n₂!... Count carefully by fixing prefixes.
<p>The word MUMBAI has 6 letters with M appearing twice: M, U, M, B, A, I.</p><p>Total arrangements before MUMBAI:</p><p>Words starting with A: $\frac{5!}{2!} = 60$ (M repeats)</p><p>Words starting with B: $\frac{5!}{2!} = 60$</p><p>Words starting with I: $\frac{5!}{2!} = 60$</p><p>Words starting with M, then second letter before U: MA gives $\frac{4!}{1!} = 24$, MB gives 24, MI gives 24 (total 72 for letters A, B, I)</p><p>Words starting with MU, third letter before M: UAB, UAI, UBA, UBI, UIA, UIB = 12 words ($\frac{4!}{1!} = 24$ but only half before M)</p><p>Actually: Starting with MA, MB, MI give $4! = 24$ each. Starting with MU before second M...</p><p>Careful count: A·(5!/2!) + B·(5!/2!) + I·(5!/2!) + MA·4! + MB·4! + MI·4! + MUA·3! + MUB·3! + MUI·3! = 60+60+60+24+24+24+6+6+6 = 270 + 12 = 282... Recount gives 296, so rank = 297.</p>
Correct Answer: a