Let $h$:$xY$ be an unkeyed hash function. Let $|x|=\{0,1{\}}^n,|Y|=\{0,1{\}}^m.$ Suppose $c$ is a fixed constant. Suppose that each yinY has $c$ number of preimages of $h.$

a$.$ Express $c$ in terms of $n$ and $m.$

b$.$ Suppose that we have only oracle access to $h.$ Given $x_0$inx, you decide to choose three distinct vectors $x_1,x_2,x_{3}in\frac{x}{\frac{?}{?}}\{x_0\}$ randomly. What is the probability that $h(x_1)=h(x_2)=h(x_3)=h(x_0).$

c$.$ If we choose two distinct vectors uniformly at random from $x.$ What is the probability of finding a collision?