The number of ways in which we can select three numbers from $1$ to $30$ so as to exclude every selection of all even numbers is

If $a_n = \sum_{r = 0}^n \frac{1}{^nC_r}$,then $\sum_{r = 0}^n \frac{r}{^nC_r}$ equals

There are three bags $B_1$,$B_2$,and $B_3$ containing $2$ Red and $3$ White,$5$ Red and $5$ White,and $3$ Red and $2$ White balls respectively. $A$ ball is drawn from bag $B_1$ and placed in bag $B_2$,then a ball is drawn from bag $B_2$ and placed in bag $B_3$,then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed,if the same colour balls are used in the first and second transfers (Assume all balls to be different) is

When ten persons shake hands with one another,in how many ways is it possible?

The sum of the digits in the unit's place of all the $4-$digit numbers formed by using the digits $3, 4, 5,$ and $6$,without repetition,is

How many numbers can be formed from the digits $1, 2, 3, 4$ when the repetition is not allowed?