Six objects $O_1$ to $O_6$ are arranged one on top of the other. In how many ways can these be arranged such that $O_1$ and $O_2$ are the $2$ bottom-most objects?

$5$-digit numbers are to be formed using $2, 3, 5, 7, 9$ without repeating the digits. If $p$ is the number of such numbers that exceed $20000$ and $q$ is the number of those that lie between $30000$ and $90000$,then $p : q$ is

The number of ways in which $n$ boys and $n$ girls can be arranged in a row such that all the boys are together and all the girls are also together is equal to

Find the number of words with or without meaning which can be made using all the letters of the word $AGAIN$. If these words are written as in a dictionary,what will be the $50^{\text{th}}$ word?

How many numbers can be formed using the digits $0, 1, 2, 3, 4, 5$ that are greater than $3000$ (repetition is not allowed)?

How many words,with or without meaning,each of $3$ vowels and $2$ consonants can be formed from the letters of the word $INVOLUTE$?