Find the probability that in a random arrangement of the letters of the word 'UNIVERSITY' the two I's do not come together ?

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Solution

$T o t a l n u m b e r o f w o r d s f o r m e d b y U N I V E R S I T Y = \frac{10!}{2!} T a k i n g t w o I' s t o g e t h e r, n u m b e r o f w a y s t o a r r a n g e l e t t e r s o f t h e w o r d U N I V E R S I T Y = 9! P r o b a b i l i t y (t w o I' s c o m e t o g e t h e r) = \frac{9!}{\frac{10!}{2!}} = \frac{1}{5} s o, P r o b a b i l i t y (t w o I' s d o n o t c o m e t o g e t h e r) = 1 - \frac{1}{5} = \frac{4}{5}$

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