The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} \cdot {}^{23}C_{r}$ is $.......$

Ten students are seated at random in a row. The probability that two particular students are not seated side by side is

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

$A$ debate club consists of $6$ girls and $4$ boys. $A$ team of $4$ members is to be selected from this club,including the selection of a captain (from among these $4$ members) for the team. If the team has to include at most one boy,then the number of ways of selecting the team is

In how many ways can $3n$ distinct objects be distributed equally among $3$ persons?

Let $n \geq 3$ be an integer. For a permutation $\sigma = (a_1, a_2, \ldots, a_n)$ of $(1, 2, \ldots, n)$,we define $f_\sigma(x) = a_n x^{n-1} + a_{n-1} x^{n-2} + \ldots + a_2 x + a_1$. Let $S_\sigma$ be the sum of the roots of $f_\sigma(x) = 0$ and let $S$ denote the sum over all permutations $\sigma$ of $(1, 2, \ldots, n)$ of the values $S_\sigma$. Then,