$\binom{n}{r+1} + 2\binom{n}{r} + \binom{n}{r-1} = \dots$

The value of $r$ for which $^{15}C_{r+3} = ^{15}C_{2r-6}$ is

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

If $^{20}C_{n+2} = ^nC_{16}$,then the value of $n$ is

Let $S = \{1, 2, 3, \ldots, 9\}$. For $k = 1, 2, \ldots, 5$,let $N_k$ be the number of subsets of $S$,each containing five elements out of which exactly $k$ are odd. Then $N_1 + N_2 + N_3 + N_4 + N_5 =$

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then: