There are $10$ points in a row. In how many ways can $4$ points be selected such that no two of them are consecutive?

$3$ boys and $3$ girls are to be seated around a circular table. Among the boys,$X$ does not want any girl as a neighbour,and among the girls,$Y$ does not want any boy as a neighbour. How many such arrangements are possible?

The number of four-digit numbers strictly greater than $4321$ that can be formed using the digits $0, 1, 2, 3, 4, 5$ (repetition of digits is allowed) is

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

The number of straight lines that can be formed by joining $20$ points,no three of which are in the same straight line,except $4$ of them which are in the same line.

In how many ways can the letters of the word '$UNIVERSAL$' be arranged? In how many of these will $E, R, S$ always occur together?