Out of $10$ persons,$A, B$,and $C$ are to speak at a function. If $A$ wants to speak before $B$ and $B$ wants to speak before $C$,in how many ways can they speak?

The sum of all the $4$-digit distinct numbers that can be formed with the digits $1, 2, 2,$ and $3$ is:

Find the number of ways of arranging $6$ red balls and $6$ black balls in a row such that no two black balls are together.

$A$ three-digit number $n$ is such that the last two digits are equal and differ from the first digit. The number of such $n$'s is:

Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First,the women choose chairs from amongst the chairs marked $1$ to $4$,and then the men select the chairs from amongst the remaining. The number of possible arrangements is

How many $6$-digit numbers can be formed from the digits $0, 1, 3, 5, 7,$ and $9$ which are divisible by $10$,given that no digit is repeated?