$^nC_{r+1} + ^nC_{r-1} + 2^nC_r = \dots$

The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples,is

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

If $^{n-1}C_r = (k^2 - 3) \cdot ^nC_{r+1}$,then an interval containing the values of $k$ is

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is

If a team of $4$ persons is to be selected out of $4$ married couples to play a mixed doubles tennis game,then the number of ways of forming a team in which no married couple appears is: