If $^{n} C _{9}=\,\,^{n} C _{8},$ find $^{n} C _{17}$

To fill $12$ vacancies there are $25$ candidates of which five are from scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all, then the number of ways in which the selection can be made

For non-negative integers $s$ and $r$, let

$\binom{s}{r}=\left\{\begin{array}{ll}\frac{s!}{r!(s-r)!} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{array}\right.$

For positive integers $m$ and $n$, let

$(m, n) \sum_{ p =0}^{ m + n } \frac{ f ( m , n , p )}{\binom{ n + p }{ p }}$

where for any nonnegative integer $p$,

$f(m, n, p)=\sum_{i=0}^{ p }\binom{m}{i}\binom{n+i}{p}\binom{p+n}{p-i}$

Then which of the following statements is/are $TRUE$?

$(A)$ $(m, n)=g(n, m)$ for all positive integers $m, n$

$(B)$ $(m, n+1)=g(m+1, n)$ for all positive integers $m, n$

$(C)$ $(2 m, 2 n)=2 g(m, n)$ for all positive integers $m, n$

$(D)$ $(2 m, 2 n)=(g(m, n))^2$ for all positive integers $m, n$

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw

If $^n{C_3} + {\,^n}{C_4} > {\,^{n + 1}}{C_3},$ then

If $^n{C_{r - 1}} = 36,{\;^n}{C_r} = 84$ and $^n{C_{r + 1}} = 126$, then the value of $r$ is