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  4. Words from the letters of the word PROBABILITY are formed by taking all letters at a time. The probability that both B 's are not together and both I 's are not together is

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Q. Words from the letters of the word PROBABILITY are formed by taking all letters at a time. The probability that both $B$ 's are not together and both $I$ 's are not together is

Probability

Solution:

The total number of cases is $11 ! / 2 ! \times 2 !$. The number of favorable cases is $[11 ! /(2 ! \times 2 !)]-9 !$ Therefore, the required probability is
$1-\frac{9 ! \times 4}{11 !}=\frac{53}{55}$