From a group of $10$ men and $8$ women,the number of ways of forming a committee of $8$ members with not more than $5$ men and not less than $5$ women is

If $^{n + 1}C_3 = 2 \cdot ^nC_2$,then $n =$

There are ten boys $B_{1}, B_{2}, \ldots, B_{10}$ and five girls $G_{1}, G_{2}, \ldots, G_{5}$ in a class. The number of ways of forming a group consisting of three boys and three girls,such that $B_{1}$ and $B_{2}$ are not both members of the same group,is

The value of ${ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4$ is

$^{n-1}C_3 + ^{n-1}C_4 > ^nC_3$,then the value of $n$ is

$^{47}C_4 + \sum_{r=1}^5 {}^{52-r}C_3 = $