$A$ student is asked to answer $10$ out of $13$ questions in an examination such that he must answer at least four questions from the first five questions. Then the total number of possible choices available to him is

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men such that it contains at least $5$ women. Find the number of ways the committee can be formed such that women are in the majority and the number of ways the committee can be formed such that men are in the majority,respectively.

$A$ box contains $2$ white balls,$3$ black balls,and $4$ red balls. In how many ways can $3$ balls be drawn from the box if at least one black ball is to be included in the draw?

Consider three boxes,each containing $10$ balls labelled $1, 2, \dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is:

The number of ways in which $3$ identical balls can be distributed into $7$ distinct bins is

$A$ country has $10$ smart cities. The government decides to connect all these cities by road. How many roads does the government need to construct so that every city is connected to every other city?