If $\binom{n+1}{3} = 2 \times \binom{n}{2}$,then $n = \dots$

If ${}^{10}P_r = 604800$ and ${}^{10}C_r = 120$,then $r =$

If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ 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 $N_1 + N_2 + N_3 + N_4 + N_5 =$

At an election,a voter may vote for any number of candidates not exceeding the number to be elected. If $4$ candidates are to be elected out of the $12$ contested in the election and a voter votes for at least one candidate,then the number of ways in which a voter can vote is:

An urn contains $5$ red marbles,$4$ black marbles and $3$ white marbles. The number of ways in which $4$ marbles can be drawn so that at most three of them are red is