Question $8)$ Given $n$ and $k$ with $nk0,$ we want to compute the $"n$ Choose $k"$ function

defined by: $(10$ points$)$

$C(n,k)=\frac{n!}{r{!}^{**}(n-k)!}$

However, we don't want to use equation $(1)$ to compute the $"$n Choose $k",$ since $(1)$ has a

factorial function that is hard to compute.

So$,$ we define the following recursive method to compute $C(n,k)$ for any $n$ and $k$ :

Recursive Function:

$C(n,k)=C(n-1,k-1)+C(n-1,k)$

for $n>k>0$

Base case:

$C(n,0)=1$ and $C(n,n)=1$

for $n0$

$($a$)$ Compute $C(5,3)$ using the above recurrence.

$($b$)$ Give pseudocode for a dynamic programming algorithm to compute $C(n,k)$ using

the above recurrence.

$($c$)$ Show the dynamic table $($memory$)$ your algorithm creates to compute $C(5,3).$