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  • the number of arrangements of the letters of the word VEER BHADUR in which two vowels are together and two vowels are together but separated

The number of arrangements of the letters of the word ′VEER BHADUR′in which two vowels are together and other two vowels are together but separated from each other-

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\\\text{VEER BHADUR }\\ (\mathrm{EE})(\mathrm{AU})$ and \\ (EU) (AE) are the only pair of Vowels possible. VRBHDR can be arranged in $\frac{6 !}{2}$ ways $(\mathrm{R}$ is repeated $)-\ldots-(1)$ \\ The vowel pairs can be placed in between the spaces \\ of the alphabets from (1) \\ For (EE) (AU) $\rightarrow$ 2 arrangements \\ (EU) (AE) $\rightarrow 4$ arrangements \\ Number of arrangements

\large \begin{array}{l} =\frac{6 !}{2}\left[7 p_{2} \times 2+7 p_{2} \times 4\right] \\\\ =\frac{6 !}{2}[84+168] \\\\ =\frac{7 !}{2}[12+24]=\frac{36 \times 7 !}{2} \\\\ =18 \times 7 ! \text { ways. } \end{array}

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