If $N(n) = n \prod_{r=1}^{2023} (n^2 - r^2)$ for $n > 2023$,then find the value of ${}^{N}C_{N-1}$ when $n = 2024$.

$A$ village has $10$ players. $A$ team of $6$ players is to be formed. $5$ members are chosen out of these $10$ players and then the captain is chosen from the remaining $5$ players. The total number of ways of choosing such a team is:

$A$ person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$,then $n$ equals

If $\binom{18}{15} + 2\binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$,then find the value of $n$.

$\binom{n}{r+1} + 2\binom{n}{r} + \binom{n}{r-1} = \dots$

In how many ways can a committee of $6$ members be formed from $8$ men and $4$ women such that the committee contains at least $3$ women?