Question

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

 

Answer 1

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