There are $5$ letters and $5$ directed envelopes. Find the number of ways in which the letters can be put into the envelopes so that all are not put in directed envelopes?

The sum of the digits in the unit place of all numbers formed with the help of $3, 4, 5, 6$ taken all at a time is

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

In how many ways can $n$ books be arranged in a row so that two specified books are not together?

In a conference of $8$ persons,if each person shakes hands with every other person exactly once,then the total number of handshakes is:

If $33!$ is divisible by $2^n$,where $n \in N$,then the sum of all possible values of $n$ is: