In how many different ways can the letters of the word "MATHEMATICS" be arranged such that consonants occur together?

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$$\begin{gathered} \mathrm{We} \mathrm{have} \mathrm{MATHEMATICS} \\ \mathrm{We} \mathrm{have} 2 \mathrm{A}\text{'}\mathrm{s}, 1 \mathrm{C},1 \mathrm{E}, 1 \mathrm{H},1\mathrm{I}, 2 \mathrm{M}\text{'}\mathrm{s}, 1 \mathrm{S},2\mathrm{T}\text{'}\mathrm{s}=11 \mathrm{letters} \\ \mathrm{Out} \mathrm{of} \mathrm{total} \mathrm{arrangents} \mathrm{we} \mathrm{will} \mathrm{have} \mathrm{to} \mathrm{divide} \mathrm{arrangements} \mathrm{of} \mathrm{letters} \mathrm{which} \mathrm{are} \mathrm{occuring} \mathrm{twice} \\ \mathrm{to} \mathrm{compensate} \mathrm{for} \mathrm{the} \mathrm{overcounting}. \\ \mathrm{Total} \mathrm{consonants}=1+1+2+1+2=7 \\ \mathrm{We} \mathrm{will} \mathrm{tie} 7 \mathrm{consonants} \mathrm{together} \mathrm{so} \mathrm{that} \mathrm{they} \mathrm{are} \mathrm{always} \mathrm{together} \mathrm{and} \mathrm{arrange} \mathrm{them} \mathrm{internally}. \\ \mathrm{So} \mathrm{effectively}, \mathrm{we} \mathrm{have} 5 \mathrm{letters} \mathrm{now}. \left(4 \mathrm{vowels}+7 \mathrm{consonants} \mathrm{tied} \mathrm{together}\right) \\ \Rightarrow \mathrm{Number} \mathrm{of} \mathrm{arrangements}=\frac{5! 7!}{2! 2! 2!} \left[\mathrm{Here} 7! \mathrm{is} \mathrm{for} \mathrm{internal} \mathrm{arrangements} \mathrm{of} \mathrm{consonants}\right] \\ =\frac{5.4.3!.7.6.5!}{2.2.2} \\ =5.3!.7.3.5! \\ =5.6.7.3.120 \\ \mathbf{\Rightarrow }\mathbf{Number}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{arrangements}\mathbf{=}\mathbf{75600}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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