How many $3$-digit numbers can be formed by using the digits $1, 3, 6,$ and $8$,when the digits may be repeated any number of times?

In how many different ways can the letters of the word '$DETAIL$' be arranged in such a way that the vowels occupy only the odd positions?

The number of ways in which any four letters can be selected from the word $CORGOO$ is

There are $n$ white and $n$ black balls marked $1, 2, 3, ..., n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours are

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

The number of ways in which the letters of the word $ARRANGE$ can be arranged such that both $R$ do not come together is