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:

Let $n$ be the number of different $5$-digit numbers,divisible by $4$,formed with the digits $1, 2, 3, 4, 5,$ and $6$,such that no digit is repeated in the numbers. What is the value of $n$?

If all the words,with or without meaning,are written using the letters of the word $QUEEN$ and are arranged as in an English dictionary,then the position of the word $QUEEN$ is (in $^{th}$)

In how many different ways can the letters of the word '$PRAISE$' be arranged?

Let $S$ be the set of all triangles in the $xy$-plane,each having one vertex at the origin and the other two vertices lying on the coordinate axes with integral coordinates. If each triangle in $S$ has an area of $50$ sq. units,then the number of elements in the set $S$ is:

How many triangles can be formed by joining four points on a circle?