Question

The number of ways of arranging the letters $AAAAA,BBB,CCC,D,EE,F$ in a row when no two $C$ 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

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

Total number of letters =15, total number of C's =3.

First, place 12 letters other than C's at dot places(.) X.X.X.X.X.X.X.X.X.X.X.X.X

The number of ways $=\frac{12!}{5!2!3!}$

Since no 2 C's are together, Place C's at cross places (X) whose number =13

their arrangements are in $\frac{{{}^{13}P}_3}{3!}$ ways.

Thus total number of ways $=\frac{12!}{5!2!3!}\times \frac{{{}^{13}P}_3}{3!}$