In how many different ways can the letters of the word $TABLE$ be arranged?

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?

There are $4$ parcels and $5$ post-offices. In how many different ways can the registration of the parcels be made?

$A$ country has $10$ smart cities. The government decides to connect all these cities by road. How many roads does the government need to construct so that every city is connected to every other city?

$A$ library has $a$ copies of one book,$b$ copies of each of two books,$c$ copies of each of three books,and single copies of $d$ books. The total number of ways in which these books can be arranged is

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?