If ${}^nC_4, {}^nC_5$ and ${}^nC_6$ are in $A.P.$,then $n$ is

If $10 \cdot ^nC_2 = 3 \cdot ^{n+1}C_3$,then the value of $n$ is

If ${ }^9 C_3+{ }^9 C_5={ }^{10} C_r$ for some $r \in N$,then $r=$

Consider three boxes,each containing $10$ balls labelled $1, 2, \dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is:

The least value of natural number $n$ satisfying $C(n, 5) + C(n, 6) > C(n + 1, 5)$ is

In how many ways can $16$ rupees be divided among $4$ persons such that no person receives less than $3$ rupees?