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?

The number of four-digit numbers strictly greater than $4321$ that can be formed using the digits $0, 1, 2, 3, 4, 5$ (repetition of digits is allowed) is

In a plane there are $37$ straight lines of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides,no three lines pass through one point,no line passes through both points $A$ and $B$,and no two lines are parallel. Then the number of intersection points the lines have is equal to

How many $3$-digit numbers,each less than $600$,can be formed from the digits $1, 2, 3, 4, 5,$ and $9$,if repetition of digits is allowed?

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84,$ then the value of $m$ is

In how many ways can a student choose a programme of $5$ courses if $9$ courses are available and $2$ courses are compulsory for every student?