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

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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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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Text Solution

Solution : There are `10` letters in the word UNIVERSITY.<br> So, there are total `(10!)/(2!)` ways of arranging these letters as `I` comes twice in the word.<br> Now, we have to find number of ways that the two `I` come together.<br> We can consider these two `I` as a single entity as they will always be together.<br> So, we can arrange the word in `9!` ways.<br> `:.` The probability two `I` are together `= (9!)/((10!)/(2!)) = 2/10 = 1/5.`<br> If we want to calculate the probability so that two I are not together, it will be ` = 1-1/5 = 4/5.`