Question

The number of ways of arranging the letters AAAAA , BBB , D , EE & F in a row if the letter C are separated from one another is

• A. ${}^{13}{c}_3.\frac{12!}{5!3!2!}$
• B. $\frac{13!}{5!3!2!}$
• C. $\frac{14!}{3!,32!}$
• D. none

Answer 1

The correct option is A ${}^{13}{c}_3.\frac{12!}{5!3!2!}$

solve:-

$\begin{array}{c}A\to 5\\ B\to 3\\ C\to 3\\ D\to 1\\ E\to 2\\ F\to 1\end{array}$

Then arrange $C$ in three ways,

$C{}^{\downarrow }C{}^{\downarrow }C$

Probability of $C$ arrange

$\underset{1}{|}-\underset{2}{|}-\underset{3}{|}-\underset{4}{|}-\underset{5}{|}-\underset{6}{|}-\underset{7}{|}-\underset{8}{|}-\underset{9}{|}-\underset{10}{|}-\underset{11}{|}-\underset{12}{|}-\underset{13}{|}$

$13$ ways to place in $C$

${}^{13}{C}_3\cdot \frac{12!}{5!\cdot 3!\cdot 2!}$

$\frac{13\times 12\times 11}{3!}\cdot \frac{12!}{5!\cdot 3!\cdot 2!}\left(\begin{array}{c}Where\\ 3!=6\\ >12\\ 2!=2\end{array}\right)$

$11\cdot \frac{13!}{5!\cdot 3!}Or11\cdot \frac{13!}{6!}$

Hence,

Option $\left(A\right)$ is correct answer