Question

Find the number of ways of arranging the letters
$AAAAA$ $BBB$ $CCC$ $D$ $EE$ $F$
in row if the letter $C$ are seperated from one another.

Answer 1

Let us ignore $3C^s$ and thus we have $12$ letters $(5A^s,3B^s,2E^s,1D,1F)$ and these can be arranged in
$\frac{12!}{5!3!2!}$ ways. ....(1)
Now after arranging these $12$ letters $(No.C^s)$ these will be $13$ gaps as in which $3$ different letters are alike, the number of distinct ways will be
$\frac{1}{3!}{(}^{13}{P}_3)=\frac{1}{6}\cdot \frac{13!}{10!}$ ....(2)
$\therefore$ The total number of words in which $C^s$ are separated from one another is
$\frac{12!}{5!\times 3!\times 2!}\times \frac{1}{6}\cdot \frac{13!}{10!}$
$=\frac{12\times 11\times 10\times 9\times 8\times 7\times 6}{6\times 2}\times \frac{1}{6}(13\times 12\times 11)$
$=95135040$