What is the probability of the random arrangement of letter in the word university and two eyes should come together?

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KR

Patna women's college

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

Out of the letters in the word ‘UNIVERSITY’, there are two I’s.
Number of permutations = \[\frac{10!}{2}\]

The number of words in which two I’s are never together is given by
total number of words – number of words in which two I’s are together.

\[= \frac{10!}{2} - 9!\]\[ = \frac{10! - 2 \times 9!}{2}\]\[ = \frac{9!\left[ 10 - 2 \right]}{2}\]\[ = \frac{9! \times 8}{2}\]

\[ = 9! \times 4\]

∴ Required probability =\[\frac{9! \times 4}{\frac{10!}{2}} = \frac{9! \times 4 \times 2}{10 \times 9!} = \frac{8}{10} = \frac{4}{5}\]

Concept: Random Experiments

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