$\mathop \sum \limits_{0 \le i < j \le n} i\left( \begin{array}{l}
n\\
j
\end{array} \right)$ is equal to
$n^22^{n-1}$
$(n^2 -1)2^{n-1}$
$(n-1)^22^n$
$n(n-1)2^{n-3}$
In a conference of $8$ persons, if each person shake hand with the other one only, then the total number of shake hands shall be
The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is
In how many ways can a girl and a boy be selected from a group of $15$ boys and $8 $ girls
How many numbers greater than $1000000$ can be formed by using the digits $1,2,0,2,4,2,4 ?$
A group of $9$ students, $s 1, s 2, \ldots, s 9$, is to be divided to form three teams $X, Y$ and, $Z$ of sizes $2,3$ , and $4$, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is. . . .