Question

The number of ways of arranging the letters $AAAAA,BBB,CCC,D,EE$ & $F$ in a row if no two ${}^{\prime }{C}^{\prime }s$ are together:

• A. ${}^{13}{C}_3\cdot \frac{12!}{5!3!2!}$
• B. $\frac{13!}{5!3!3!2!}$
• C. $\frac{14!}{5!3!2!}$
• D. $11.\frac{13!}{6!}$

Answer 1

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

Total number of letters $=15,$
number of $C^{\prime }s=3$
First place $12$ letter other then $C^{\prime }s$ at dot places
$X.X.X.X.X.X.X.X.X.X.X.X.X$
The number of ways $=\frac{12!}{5!3!2!}$
Since no two $C^{\prime }s$ are together
Therefore place $C^{\prime }s$ at cross places whose number $=13$
This arrangements are ${}^{13}{P}_3$
The total number of ways $=\frac{12!}{5!3!2!}\times \frac{{}^{13}{P}_3}{3!}{=}^{13}{C}_3\frac{12!}{5!3!2!}$