The number of ways in which a committee of $7$ members can be formed from $6$ teachers,$5$ fathers,and $4$ students in such a way that at least one from each group is included and teachers form the majority among them,is

The remainder of $n^4-2n^3-n^2+2n-26$ when divided by $24$ is:

Let $x$ denote the number of ways of selecting at least one ball from a bag containing $3$ identical red balls,$4$ identical blue balls,and $5$ identical green balls. Let $y$ denote the number of ways in which a student can fail in an examination,when they have to write the examination in $5$ different subjects. Then $x+y=$

The number of $4$ letter words (with or without meaning) that can be formed from the eleven letters of the word $'EXAMINATION'$ is

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