Out of $6$ books,in how many ways can a set of one or more books be chosen?

There are $10$ persons named $A, B, \dots, J$. We have the capacity to accommodate only $5$. In how many ways can we arrange them in a line if $A$ must be included and $G$ and $H$ must not be included in the team of $5$?

The number of $3$-digit odd numbers that can be formed by using the digits $1, 2, 3, 4, 5, 6$ when the repetition is allowed,is

The total number of seven-digit numbers the sum of whose digits is even is

There are $m$ points on a straight line $AB$ and $n$ points on another line $AC$,none of them being the point $A$. Triangles are formed from these points as vertices when $(i)$ $A$ is excluded and $(ii)$ $A$ is included. Then the ratio of the number of triangles in the two cases is

The number of ways that $8$ beads of different colours can be strung as a necklace is: