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In a random arrangement of the letters of the word 'UNIVERSITY', what is the probability that two I's do not come together?
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$4 / 5$
Total number of words formed byletters of UNIVERSITY
$=\frac{10 !}{2 !} \quad(\because \mathrm{I}$ is repeated $)$
Taking two $\mathrm{I}_{\mathrm{s}}$ together, number of ways to arrange letters of UNIVERSITY $=9 !$ $\therefore$ Probability of two $\mathrm{I}_{\mathrm{s}}$ coming together
$=\frac{9 !}{\frac{10 !}{2 !}}=\frac{9 ! \times 2 !}{10 !}=\frac{2}{10}=\frac{1}{5}$
$\therefore$ Probability of two $\mathrm{I}_{\mathrm{s}}$ not coming together
$=1-\frac{1}{5}=\frac{4}{5}$
$=\frac{10 !}{2 !} \quad(\because \mathrm{I}$ is repeated $)$
Taking two $\mathrm{I}_{\mathrm{s}}$ together, number of ways to arrange letters of UNIVERSITY $=9 !$ $\therefore$ Probability of two $\mathrm{I}_{\mathrm{s}}$ coming together
$=\frac{9 !}{\frac{10 !}{2 !}}=\frac{9 ! \times 2 !}{10 !}=\frac{2}{10}=\frac{1}{5}$
$\therefore$ Probability of two $\mathrm{I}_{\mathrm{s}}$ not coming together
$=1-\frac{1}{5}=\frac{4}{5}$
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