When all the students in a school are made to stand in rows of $54,$ $30$ such rows are formed. If the students are made to stand in rows of $45,$ how many such rows will be formed?

Number of ordered pairs $(x, y)$ of integers such that their product is a positive integer less than $100$ is -

If $^8C_r = ^8C_{r+2}$,then the value of $^rC_2$ is

In how many ways is it possible to choose a white square and a black square on a chess board so that the squares do not lie in the same row or column?

In how many different ways can the letters of the word $ALGEBRA$ be arranged in a row if:
$(I)$ The $2$ $A$s are together?
$(II)$ The $2$ $A$s are not together?

There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls,one in each box,could be placed such that a ball does not go to the box of its own colour is