Question

The number of ways of arranging the letter AAAAA BBB CCC D EE F in a row then no two $C^{\prime }s$ are together is

• A. $\frac{15!}{5!3!3!2!}-3!$
• B. $\frac{15!}{5!3!3!2!}-\frac{13!}{5!3!2!}$
• C. $\frac{12!}{5!3!2!}\times \frac{{}^{13}{P}_3}{3!}$
• D. $\frac{12!}{5!3!2!}{\times }^{13}{P}_3$

Answer 1

Answer: (C)

Total no. of letters $=15$. No. of $C’s=3$.
First place $12$ letters other than $C’s$ at dot places.
$\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times .\times$
The no. of ways $=\frac{12!}{5!3!2!}$
Since no two C's are together.
$\therefore$ Place C’s at cross places whose number $=13$.
Their arrangements are $\frac{{}^{13}{P}_3}{3!}$
Total no. of ways $=\frac{12!}{5!3!2!}\cdot \frac{{}^{13}{P}_3}{3!}$