Question

The number of ways of arranging the letters $AAAAA,BBB,CCC<D,EE$ and $F$ in a row if the letters $C$ are separated from one another is:

• A. ${{}^{13}C}_3\frac{12!}{5!3!2!}$
• B. $\frac{13!}{5!3!3!2!}$
• C. $\frac{14!}{5!3!2!}$
• D. $\frac{15!}{3!{(3!)}^2!}-\frac{13!}{5!3!2!}-\frac{12!}{5!3!}{{}^{13}C}_2$

Answer 1

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

$AAAAABBBCCCDEEF$

To separate all $C$,

_A_A_A_A_A_B_B_B_D_E_E_F_

Insert $C$ in between letters

No. of ways of arranging remaining letters=$\frac{12!}{5!3!2!}$$C{=}^{13}{C}_3\frac{3!}{3!}{=}^{13}{C}_3$

$\therefore$Total no. of ways${=}^{13}{C}_3\frac{12!}{5!3!2!}$