Question

If the letters of the word $RANDOM$ be arranged at random, the probability that there are exactly $2$ letters in between $A$ and $O$ is

• A. $\frac{4}{5}$
• B. $\frac{3}{5}$
• C. $\frac{2}{5}$
• D. $\frac{1}{5}$

Answer 1

The correct option is D $\frac{1}{5}$

No. of ways of putting 2 letters between A and O and then arranging the letters ${=}^4{C}_2\ast 2!\ast 2!\ast 3!=144$
(the first 2! is for A and O, the second 2! is for the letters between A and O, 3! is for the three things to be arranged, i.e. O,A and the 2 letters form a group followed by 2 other letters outside)
Probability $=\frac{144}{6!}=\frac{1}{5}$