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