Connie and Nora went into Ready Scoop to get ice cream cones, but Nora cannot make up her mind. They have $23$ flavors and she wants $3$ scoops.  

1. If Nora is very particular about the order of the scoops, how many choices does she have if all of the scoops are different?

   Hint (a):

   No repeats, but order DOES matter.

2. Nora changes her mind. She wants a dish, not a cone, but she still wants three different flavors. How many ways can she order?

   Hint (b):

   No repeats, but order DOES NOT matter.

3. Connie says, “I still want a cone with dark chocolate on the bottom and then any other two scoops.” How many cones are possible with dark chocolate on the bottom?

   Hint (c):

   Assume that repeats are acceptable $\frac{1}{\text{bottom}}\cdot\frac{22}{\text{middle}}\cdot\frac{22}{\text{top}}$

4. Vlad came in as they were leaving and saw Connie’s cone. He said, “Oh, that’s what I want, a cone with chocolate on the bottom and then two other flavors.” The clerk, said, “Okay, but we have four kinds of chocolate.” Vlad replied, “Any kind of chocolate will do.” How many different cones could fill Vlad’s order?

   Hint (d):

   See (c). Remember that Vlad has $4$ choices of chocolate for the bottom and not just one.