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 permutation $=\frac{10!}{2!}=\frac{10!}{2}$

The number os 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

$=\frac{10!}{2}-9!=9!-4$

therefore required probability $=\frac{9!-4}{\frac{10!}{2}}=\frac{4}{5}$