How many arrangements can be made of the letters of the word '$ARRANGEMENT$'?

If $P(56, r+6) : P(54, r+3) = 30800 : 1$,find $r$.

The number of times the digit $5$ will be written when listing the integers from $1$ to $1000$ is

In a shop,there are $5$ types of ice-creams available. $A$ child buys $6$ ice-creams.
Statement-$1$: The number of different ways the child can buy the $6$ ice-creams is $^{10}C_5$.
Statement-$2$: The number of different ways the child can buy the $6$ ice-creams is equal to the number of different ways of arranging $6$ $A$'s and $4$ $B$'s in a row.

How many different words can be formed with the letters of the word '$BHARAT$'? In how many of these are $B$ and $H$ never together?

The letters of the word $MODESTY$ are written in all possible orders and these words are written out as in a dictionary,then the rank of the word $MODESTY$ is