In how many ways can $6$ persons be selected from $4$ officers and $8$ constables such that at least one officer is included?

In how many ways can a committee of $5$ members be formed from $4$ gentlemen and $6$ ladies such that the number of gentlemen is greater than the number of ladies?

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

In an election,there are $5$ candidates and $3$ vacancies. $A$ voter can vote for a maximum of $3$ candidates. In how many ways can a voter cast their vote?

$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these do the four cards belong to four different suits?