If ${}^nC_4, {}^nC_5$ and ${}^nC_6$ are in $A.P.$, then $n$ is

The least value of $n$ for which ${ }^{(n-1)} C_2 + { }^{(n-1)} C_3 > { }^n C_2$ is

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

The number of $5$ card combinations out of a deck of $52$ cards such that there is exactly one ace in each combination is:

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

In how many ways can two balls of the same color be selected from $4$ distinct black balls and $3$ distinct white balls?