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.
$$\begin{gathered} =\frac{10!}{2}-9! \\ =\frac{10!-2\times 9!}{2} \\ =\frac{9!\left[10-2\right]}{2} \\ =\frac{9!\times 8}{2} \\ =9!\times 4 \end{gathered}$$
∴ Required probability = $\frac{9!\times 4}{{\displaystyle \frac{10!}{2}}}=\frac{9!\times 4\times 2}{10\times 9!}=\frac{8}{10}=\frac{4}{5}$