The number of ways in which $6$ distinct rings can be worn on the $4$ fingers of one hand is

In how many ways $n$ books can be arranged in a row so that two specified books are not together?

The range of the function $f(x) = {}^{7 - x}P_{x - 3}$ is

The number of natural numbers less than $1000$,in which no two digits are repeated,is:

There are $9$ chairs in a room on which $6$ persons are to be seated. Out of these,one person is a guest who must occupy one specific chair. In how many ways can they sit?

Words of length $10$ are formed using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated,and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then,$\frac{y}{9x} =$