The letters of the word 'QUESTION' are arranged in a row at random. The probability that there are exactly…
The letters of the word 'QUESTION' are arranged in a row at random. The probability that there are exactly two letters between $\mathrm{Q}$ and $\mathrm{S}$ is
$\frac{1}{14}$
$\frac{5}{7}$
$\frac{1}{7}$
$\frac{5}{28}$
Solution
The given word is 'QUESTION', let E be the event that there are exactly two letters between $Q$ and $S$.
Here, number of sample space,
$n(S)=8 \text { ! }$
number of possible events,
$n(E)={ }^6 P_2 \times 2 ! \times 5 \text { ! }$
$\therefore \quad$ Required probability
$\begin{aligned}
& P(E)=\frac{n(E)}{n(S)}=\frac{{ }^6 P_2 \times 2 ! \times 5 !}{8 !}=\frac{6 ! \times 2 ! \times 5 !}{4 ! \times 8 !} \\
& =\frac{6 ! \times 2 ! \times 5 ! \times 4 !}{4 ! \times 7 \times 8 \times 6 !}=\frac{2 \times 5}{7 \times 8}=\frac{5}{28}
\end{aligned}$