A country has ten smart cities. The government decides to connect all these cities by road. How many roads the government need to construct so that every city is connected to every other city ?

The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is

If $^n{C_r}$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression $^n{C_{r + 1}} + {\,^n}{C_{r - 1}} + \,2 \times {\,^n}{C_r}$ equals

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84,$ then the value of $m$ is

$\mathop \sum \limits_{0 \le i < j \le n} i\left( \begin{array}{l}
n\\
j
\end{array} \right)$ is equal to

In an examination there are three multiple choice questions and each question has $4 $ choices. Number of ways in which a student can fail to get all answers correct, is