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Q.
The probability that in a random arrangement of the letter of the word 'UNIVERSITY', the two I's do not come together is:
Permutations and Combinations
Solution:
The total number of arrangement of the letters in the word ‘UNIVERSITY’ is $\frac{10 !}{2!}$
($\because$ Total letters = 10, I occurs twice)
Now, if 2I’s occurs together, then regard 2I’s as one letter.
Then, total number of arrangement in which 2I’s are not together $ = \frac{10!}{2!} - 9!$
Hence, required prob
$= \frac{ \frac{10!}{2!} - 9!}{\frac{10!}{2!}} = 1 - \frac{9!2!}{10!} = 1 - \frac{ 1}{5} = \frac{4}{5}$