If ${ }^{18} C_{r}={ }^{18} C_{r+2}$,find ${ }^{r} C_{5}$.

There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

The number of positive integral solutions of the equation $xyz = 3000$ is:

$A$ question paper has $2$ parts,part $A$ and part $B$,each containing $10$ questions. If a student has to choose $8$ questions from part $A$ and $5$ questions from part $B$,in how many ways can the student choose the questions?

The number of natural numbers less than $7,000$ which can be formed by using the digits $0, 1, 3, 7, 9$ (repetition of digits allowed) is equal to

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)