What is the total number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are the $4$ cards of the same colour?

The number of integers between $100$ and $1000$ such that the sum of their digits is equal to $14$ is:

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 $n = 2024$.

In an election,$3$ members are to be elected from $6$ candidates. $A$ voter can cast their vote for any number of candidates but not more than the number of candidates to be elected. In how many ways can a voter cast their vote?

$A$ candidate is required to answer $6$ out of $12$ questions which are divided into two parts $A$ and $B$,each containing $6$ questions. The candidate is not permitted to attempt more than $4$ questions from any part. In how many different ways can he/she make his/her choice of $6$ questions?

If ${ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3$,then the minimum value of $n$ is