Question

In how many ways can the letter of the words $AMARAVATHI$ be arranged, so that
All the $A^{\prime }s$ are not together?

Answer 1

Total numbers of letters = 10

A occurred 4 times

total numbers of ways of letters of the word AMARAVATHI can

be arranged $=\frac{10!}{4!}=\begin{array}{c}10\times 9\times 8\times 7\times 6\times 5\end{array}$

$=151,200$

number of ways letters of the word AMARAVATHI arranged with all A's together $=7!=5040$

number of arrangement, so that

all the A's are not together = total number of ways - number of ways with all A's together

$\begin{array}{c}=151,200-5040\\ =146,160\end{array}$