In how many ways can $21$ books on Sanskrit and $19$ books on French be placed in a row on a shelf such that two books on French may not be together?

In how many ways is it possible to choose a white square and a black square on a chess board so that the squares do not lie in the same row or column?

The number of $5$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6$ such that the number must include both $1$ and $2$ is:

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

Find $n$ if ${ }^{n} P_{4}=18 \cdot{ }^{n-1} P_{2}$.

If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is