Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is

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:

In how many ways can $15$ different books be divided equally among $3$ sets or groups?

$A$ candidate is required to answer $6$ out of $10$ questions,which are divided into two groups each containing $5$ questions. The candidate is not permitted to attempt more than $4$ questions from each group. In how many ways can the candidate make their choice?

The number of ways in which a committee of $6$ members can be formed from $8$ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is:

$\sum_{0 \le i < j \le n} i \binom{n}{j}$ is equal to