A committee of $12$ is to be formed from $9$ women and $8$ men in which at least $5$ women have to be included in a committee. Then the number of committees in which the women are in majority and men are in majority are respectively

$^n{C_r}{ + ^n}{C_{r - 1}}$ is equal to

If $^{2017}C_0 + ^{2017}C_1 + ^{2017}C_2+......+ ^{2017}C_{1008} = \lambda ^2 (\lambda   > 0),$ then remainder when $\lambda $ is divided by $33$ is-

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

If $n$ is even and the value of $^n{C_r}$ is maximum, then $r = $

A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is