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

Let $n$ and $k$ be positive integers such that $n \ge \frac{k(k + 1)}{2}$. The number of solutions $(x_1, x_2, ..., x_k)$ where $x_1 \ge 1, x_2 \ge 2, ..., x_k \ge k$ are all integers,satisfying $x_1 + x_2 + ... + x_k = n$,is

The product of any $r$ consecutive natural numbers is always divisible by

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

There are $15$ players in a cricket team,out of which $6$ are bowlers,$7$ are batsmen,and $2$ are wicketkeepers. The number of ways a team of $11$ players can be selected from them so as to include at least $4$ bowlers,$5$ batsmen,and $1$ wicketkeeper 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 each child will go to the garden is