If ${}^{n-1}C_3 + {}^{n-1}C_4 > {}^{n}C_3$, then $n$ is just greater than which integer?

There are $10$ persons named $A, B, \dots, J$. We have the capacity to accommodate only $5$. In how many ways can we arrange them in a line if $A$ must be included,and $G$ and $H$ must not be included in the team of $5$?

If $\binom{n-1}{r} = (k^2 - 3) \binom{n}{r+1}$,then $k \in \dots$

Let $A_1, A_2, \dots, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, \dots, 11$. One hundred gold coins were won by the team in the final match. These coins are to be distributed among the players such that each player $A_i$ gets at least $i+1$ coins,except for the captain and vice-captain who get at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. If the captain is $A_1$ and the vice-captain is $A_2$,in how many different ways can these coins be distributed?

The number of solutions of the equation $x+y+z=10$, where $x, y$, and $z$ are positive integers, is:

If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^nC_{r-1} + ^nC_r = $