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

In the word 'UNIVERITY' there are 10 letters

$\therefore n\left(S\right)=10!$

Let E be event that both the l's come together

$n\left(E\right)=2\times 9!$

$\therefore p\left(E\right)=\frac{2\times 9!}{10!}=\frac{2}{10}=\frac{1}{5}$

$\therefore$ The probability that two l's do not come together is

$p\left(\overline{E}\right)=1-p\left(E\right)=1-\frac{1}{5}=\frac{4}{5}$

$\therefore p\left(\overline{E}\right)=\frac{4}{5}$