The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4,$ and $5$ in which at least two digits are identical is

If $11$ members of a committee sit at a round table so that the President and Secretary always sit together,then the number of arrangements is

$1 + \sum\limits_{r = 0}^{22} {\left\{ {r\left( {r + 2} \right) + 1} \right\}} \cdot r! = k!$,then the number of divisors of $k$ is

In a city,no two persons have an identical set of teeth,and there is no person without a tooth. Also,no person has more than $32$ teeth. If we disregard the shape and size of the teeth and consider only the presence or absence of a tooth at each of the $32$ positions,then the maximum population of the city is:

How many $3$-digit numbers can be formed using $0, 1, 3, 5, 7$ when repetition is allowed?

How many odd numbers of four digits can be formed with the digits $0, 1, 2, 3, 4, 7$ and $8$?