Out of $6$ books,in how many ways can a set of one or more books be chosen?

How many numbers consisting of $5$ digits can be formed in which the digits $3, 4$ and $7$ are used only once and the digit $5$ is used twice?

The number of four-letter words that can be formed from the letters of the word '$MAYANK$' such that both '$A$'s are included but never together is equal to:

The exponent of $3$ in $100!$ is

Each section in the first year of a plus $2$ course has exactly $40$ students. If there are $5$ sections,in how many ways can a set of $4$ student representatives be selected,$1$ from each section?

$A$ 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?