Evaluate $\frac{n!}{(n-r)!}$ when $n=6$ and $r=2$.

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?

The number of ways in which first,second and third prizes can be given to $5$ competitors is

How many numbers between $5000$ and $10,000$ can be formed using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ if each digit appears not more than once in each number?

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.

$A$ dictionary is printed consisting of $7$-lettered words only that can be made using the letters of the word $CRICKET$. If the words are arranged in alphabetical order,as in an ordinary dictionary,then the number of words before the word $CRICKET$ is: