Question

The number of ways of arranging the letter AAAAA BBB CCC D EE F in a row then no two $C's$ 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 $=\frac{12\,!}{5\,!\,3\,!\,2\,!}\cdot\frac{^{13}P_{3}}{3\,!}$