Let $\left(\begin{array}{l}n \\ k\end{array}\right)$ denotes ${ }^{n} C_{k}$ and $\left[\begin{array}{l} n \\ k \end{array}\right]=\left\{\begin{array}{cc}\left(\begin{array}{c} n \\ k \end{array}\right), & \text { if } 0 \leq k \leq n \\ 0, & \text { otherwise }\end{array}\right.$

If $A_{k}=\sum_{i=0}^{9}\left(\begin{array}{l}9 \\ i\end{array}\right)\left[\begin{array}{c}12 \\ 12-k+i\end{array}\right]+\sum_{i=0}^{8}\left(\begin{array}{c}8 \\ i\end{array}\right)\left[\begin{array}{c}13 \\ 13-k+i\end{array}\right]$

and $A_{4}-A_{3}=190 \mathrm{p}$, then $p$ is equal to :

In a football championship, there were played $153$ matches. Every team played one match with each other. The number of teams participating in the championship is

In how many ways a team of $11$ players can be formed out of $25$ players, if $6$ out of them are always to be included and $5$ are always to be excluded

The sum $\sum\limits_{i = 0}^m {\left( {\begin{array}{*{20}{c}}{10}\\i\end{array}} \right)} \,\left( {\begin{array}{*{20}{c}}{20}\\{m - i}\end{array}} \right)\,,$ $\left( {{\rm{where}}\,\left( {\begin{array}{*{20}{c}}p\\q\end{array}} \right)\, = 0\,{\rm{if}}\,p < q} \right)$, is maximum when m is

In an examination of Mathematics paper, there are $20$ questions of equal marks and the question paper is divided into three sections : $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$. A student is required to attempt total $15$ questions taking at least $4$ questions from each section. If section $A$ has $8$questions, section $\mathrm{B}$ has $6$ questions and section $\mathrm{C}$ has $6$ questions, then the total number of ways a student can select $15$ questions is

A father with $8$ children takes them $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once. The number of times he will go to the garden is