If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

$^{n-1}C_r = (k^2 - 8) ^nC_{r+1}$ if and only if:

If $x, y$ and $r$ are positive integers,then $^x{C_r} + ^x{C_{r-1}} ^y{C_1} + ^x{C_{r-2}} ^y{C_2} + \dots + ^y{C_r} = $

There are $10$ persons named $A, B, \dots, J$. We have the capacity to accommodate only $5$. In how many ways can we arrange them in a line if $A$ must be included,and $G$ and $H$ must not be included in the team of $5$?

$A$ group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least $3$ girls?

If there are $6$ alike fruits,$7$ alike vegetables,and $8$ alike biscuits,then the number of ways of selecting any number of things out of them such that at least one from each category is selected,is