Suppose that $20$ pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar is connected by beams to the top of all its non-adjacent pillars,then the total number of beams is:

Six '$+$' and four '$-$' signs are to be placed in a straight line so that no two '$-$' signs come together. Then,the total number of ways is:

There are seven candidates for $3$ posts. In how many ways can the posts be filled?

Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First,the women choose the chairs from amongst the chairs marked $1$ to $4$,and then men select the chairs from amongst the remaining. The number of possible arrangements is

The number of ways in which the following prizes can be given to a class of $20$ boys,first and second Mathematics,first and second Physics,first Chemistry,and first English is:

In how many ways can a cricket team of $11$ players be selected out of $16$ players,if $2$ particular players are always to be included?