In how many ways can you arrange the letters of the word "$ATTAIN$" so that the $T$'s are together?

$A$ vehicle registration number consists of $2$ letters of the English alphabet followed by $4$ digits, where the first digit is not zero. The total number of vehicles with distinct registration numbers is

We are to form different words with the letters of the word $INTEGER$. Let $m_1$ be the number of words in which $I$ and $N$ are never together and $m_2$ be the number of words which begin with $I$ and end with $R$,then $m_1/m_2$ is equal to

The number of arrangements of the letters of the word $BANANA$ in which two $N$s do not appear adjacently is

The total number of $4$-digit numbers which can be formed using the digits $1, 2, 3, 4$ without repetition such that the digit $n+1$ never immediately follows the digit $n$ is:

Find the number of $4$ letter words,with or without meaning,which can be formed out of the letters of the word $ROSE$,where the repetition of the letters is not allowed.