Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Find the Probability that in a Random Arrangement of the Letters of the Word 'University', the Two I'S Do Not Come Together. - Mathematics

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

#### Solution

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 =$\frac{9! \times 4}{\frac{10!}{2}} = \frac{9! \times 4 \times 2}{10 \times 9!} = \frac{8}{10} = \frac{4}{5}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 33 Probability
Exercise 33.3 | Q 24 | Page 47