Problem $1.[30$ points$]$ The mathematical combination function $c(n,k)$ is usually

defined in terms of factorials, as follows:

$c(n,k)=\frac{n!}{k!(n-k)!}$

$(1)$ Write a factorial function that takes a positive integer $n$ and returns $n!,$ that is

the product of all positive integers less than or equal to $n.$

$(2)$ Write another function that computes $c(n,k)$ by calling the factorial function.

$(3)$ Write a main program that asks the user to enter a positive integer. Assuming

the user always enters a valid integer number, the program should verify

that $n$ is greater than $0$ and if not, ask for another, checks again, until a

correct input was entered. The program then calls the combination function

and displays the value $c(n,k)$ for $k=1,2,$dots,$n$ on the screen.