Assuming the balls to be identical except for the difference in colours,the number of ways in which one or more balls can be selected from $10$ white,$9$ green,and $7$ black balls is:

$3$ 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?

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

The number of ways in which $10$ persons can go in two boats so that there may be $5$ on each boat,supposing that two particular persons will not go in the same boat is:

There are $3$ girls in a class of $10$ students. The number of different ways in which they can be seated in a row such that no two of the three girls are together is