Question

The number of ways of arranging the letters $\text{AAAAABBBCCCDEEF}$ in a row when no two $\text{C's}$ 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

The correct option is C

$\frac{12!}{5!3!2!}\times \frac{{}^{13}{P}_3}{3!}$

Explanation for the correct option(s)

Find the number of ways.

Consider the letters in row $\text{AAAAABBBCCCDEEF}$

The total number of letters is equal to $15$ and there are three $\text{C's}$ here.

The number of ways of arrangement for $12$ letters other than $\text{C's}$ can be expressed as$\frac{12!}{5!2!3!}$

When no two $\text{C's}$ are together, then number of arrangements of $\text{C}$ will be expressed as $\frac{{}^{13}{P}_3}{3!}$

The total number of ways are arranged as $\frac{12!}{5!3!2!}\times \frac{{}^{13}{P}_3}{3!}$

Hence, the correct answer is Option C