Out of $11$ consecutive natural numbers,if three numbers are selected at random (without repetition),then the probability that they are in $A.P.$ with a positive common difference is:

Out of $6$ distinct novels and $3$ distinct dictionaries,$4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf such that the dictionary is always in the middle. What is the number of such arrangements?

If ${}^n C_{r-1}=36$, ${}^n C_r=84$, and ${}^n C_{r+1}=126$, then the value of ${}^n C_8$ is

Let $A$ be a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen, and the set $A$ is reconstructed by replacing the elements of $P$. $A$ subset $Q$ of $A$ is chosen again. The number of ways of choosing $P$ and $Q$ such that $Q$ contains just one element more than $P$ is

How many $5$-digit numbers can be formed using the digits $0, 1, 2, 3, 4,$ and $5$ without repetition such that the number is divisible by $3$?

Three and four digit numbers are formed from the digits $1, 3, 5, 6, 8$. If $e_1$ is the number of three-digit even numbers with no digit repeated and $e_2$ is the number of four-digit even numbers with no digit repeated. Also,$O_1$ represents the number of three-digit odd numbers in which no digit is repeated and $O_2$ represents the number of four-digit odd numbers in which no digit is repeated. Then: