The arrangements of the letters of MUMBAI are listed in dictionary order. In which position does MUMBAI appear?

Sorted letters: A, B, I, M, M, U. Words starting with A, B or I: each leaves 5 letters with M twice, $\frac{5!}{2!}=60$, so 180. Starting MA, MB, MI or MM: $4!=24$ each, 96. Starting MUA, MUB, MUI: $3!=6$ each, 18. Then MUM + (A, B, I): MUMABI, MUMAIB, MUMBAI. Position $180+96+18+3=297$.