Question

If the letters of the word $QUESTION$ are arranged at random, the probability that there will be exactly $2$ letters in between $Q$ and $U$ is

• A. $\frac{{}^6{P}_2}{8!}$
• B. $\frac{{}^6{P}_2\times 5!}{8!}$
• C. $\frac{{}^6{P}_2\times 5!\times 2}{8!}$
• D. $\frac{{}^6{P}_2\times 2}{8!}$

Answer 1

The correct option is C $\frac{{}^6{P}_2\times 5!\times 2}{8!}$

$|T|I|O|N|$ and $Q--U$
We could put the set of Q and U at 5 places, and could also arrange Q and U in 2 ways
Selection of 2 letters from 6${=}^6{C}_2$
Probability that there will be exactly 2 letters in between Q and U is $=5\times \frac{{}^6{C}_2\times 2!\times 4!\times 2}{8!}$
$\frac{{}^6{P}_2\times 5!\times 2}{8!}$