In how many of the distinct permutations of the letters in the MISSISSIPPI do the four I's not come together?

In how many ways can the letters of the word PERMUTATIONS be arranged, if the words start with P and end with S?

The number of ways in which seven persons can be arranged at a round table, if two particular persons may not sit together is

How many numbers lying between $10$ and $1000$ can be formed from the digits $1,2,3,4,5,6,7,8,9$ (repetition of digits is allowed?)

Everybody in a room shakes hands with everybody else. The total number of hand shakes is $66$. The total number of persons in the room is 

A question paper is divided into two parts $A$ and $B$ and each part contains $5$ questions. The number of ways in which a candidate can answer $6$ questions selecting at least two questions from each part is