the numbers of ways in which the letters of the word TRIANGLE can be arranged such that two vowels do not occur together 

1. 1200
2. 2400
3. 14400
4. None of these

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Question}\text{'}\mathrm{s} \mathrm{language} \mathrm{is} \mathrm{ambiguous}. \mathrm{It} \mathrm{states} \mathrm{that} \mathrm{two} \mathrm{vowels} \mathrm{should} \mathrm{not} \\ \mathrm{come} \mathrm{together} \mathrm{which} \mathrm{means} \mathrm{three} \mathrm{vowels} \mathrm{can} \mathrm{come} \mathrm{together}. \\ \mathrm{We} \mathrm{have} \\ \mathrm{TRIANGLE}\to \mathrm{T},\mathrm{R},\mathrm{I},\mathrm{A},\mathrm{N},\mathrm{G},\mathrm{L},\mathrm{E}\to 8 \mathrm{letters} \mathrm{in} \mathrm{which} 3 \mathrm{are} \mathrm{vowels} \\ \mathrm{Total} \mathrm{words}=8!=8·7·6·5·4·3·2·1=40320 \\ \mathrm{Number} \mathrm{of} \mathrm{words} \mathrm{in} \mathrm{which} \mathrm{two} \mathrm{vowels} \mathrm{are} \mathrm{together}: \\ \mathrm{We} \mathrm{select} \mathrm{two} \mathrm{vowels} \mathrm{and} \mathrm{then} \mathrm{tie} \mathrm{them} \mathrm{together} \mathrm{so} \mathrm{that} \mathrm{we} \mathrm{are} \mathrm{effectively} \\ \mathrm{left} \mathrm{with} 7 \mathrm{letters} \mathrm{and} \mathrm{also} \mathrm{we} \mathrm{need} \mathrm{to} \mathrm{take} \mathrm{care} \mathrm{of} \mathrm{internal} \mathrm{arrangement} \mathrm{of} \\ \mathrm{two} \mathrm{vowels}. \\ \Rightarrow \mathrm{Number} \mathrm{of} \mathrm{words} \mathrm{in} \mathrm{which} \mathrm{two} \mathrm{vowels} \mathrm{are} \mathrm{together}{=}^3{\mathrm{C}}_2\times 7!\times 2! \\ =3\times 7\times 6\times 5\times 4\times 3\times 2\times 1\times 2 \\ =30240 \\ \mathrm{But} \mathrm{we} \mathrm{need} \mathrm{to} \mathrm{include} \mathrm{words} \mathrm{in} \mathrm{which} \mathrm{three} \mathrm{vowels} \mathrm{are} \mathrm{together} \\ \mathrm{Number} \mathrm{of} \mathrm{words} \mathrm{in} \mathrm{which} \mathrm{three} \mathrm{vowels} \mathrm{are} \mathrm{together}{=}^3{\mathrm{C}}_3\times 6!\times 3! \\ =1\times 6\times 5\times 4\times 3\times 2\times 1\times 6 \\ =4320 \\ \Rightarrow \mathrm{Required} \mathrm{number} \mathrm{of} \mathrm{words} \\ =40320-30240+4320 \\ \mathbf{=}\mathbf{14400}\left(\mathbf{Answer}\right) \end{gathered}$$

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