The number of ways in which $5$ beads of different colours form a necklace is

Eleven books consisting of $5$ Mathematics,$4$ Physics,and $2$ Chemistry are placed on a shelf. The number of possible ways of arranging them on the assumption that the books of the same subject are all together is

How many integers between $1000$ and $10000$ have no digits other than $4, 5,$ or $6$?

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

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

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: