Ariel, Ollie and Eden are going trick-or-treating for Halloween, but first their parents are taking them to a costume store to pick out costumes. The store has 9 different costume options available – witch, pirate, cat, superhero, princess, vampire, fire fighter, zombie, clown. In how many different ways can Ariel, Ollie and Eden each select one of these costumes, with no two of them wearing the same costume?

*There are 9 options for the first costume. Since no two of them will wear the same costume, this leaves 8 options for the second costume and 7 options for the third costume. In the end, there will be 9 × 8 × 7 = ***504 ***different ways for them to each select one of the 9 costumes, with no two of them wearing the same costume. *

To carry their candy, Ariel, Ollie and Eden each have a bucket that looks like a pumpkin. It is completely spherical in shape with a straight slice across the top of the bucket that creates a circular opening for dropping in candy. If the radius of the bucket is 6 inches and the height, measured from the base of the bucket to the open top, is 9 inches, what is the area of the circular opening of the bucket, in square inches? Express your answer in terms of π.

*The spherical bucket has radius 6 inches, and the circular opening is created by a slice through the sphere 9 inches from its base. Since both measurements are with respect to the sphere’s center, let’s simplify the problem by looking at it in two dimensions. We’ll focus on a circle with points on the sphere whose center coincides with the center of the sphere, sometimes called a “great circle”. Consider the circle shown, with radius 6 inches and segment AB drawn through center P perpendicular to chord CD, such that AB = 9 inches. Notice that the chord CD is a diameter of the circular opening of the bucket. To calculate the area, we need the radius of the opening, or CB. Since AP = 6 inches, BP = 9 – 6 = 3 inches. If we draw radius CP, we create 30-60-90 right triangle PBC. We can conclude that CB = 3√3 inches. Therefore, the circular opening of the bucket has area π(3√3) ^{2} = *

**27π***in*

^{2}.Last year, after Ariel, Ollie and Eden went trick-or-treating, they combined all their candy together and sorted it. They found that 1/3 was chocolate candy, 1/4 was hard candy, 1/5 was candy corn and the rest were fruit chews. If they had a 180 pieces of candy total, how many pieces were fruit chews?

*If 1/3 of their candy was chocolate, 1/4 was hard candy and 1/5 was candy corn, then 1/3 + 1/4 + 1/5 = 47/60 of their candy was one of these three types. The fraction of the candy that was fruit chews would have been 1 – 47/60 = 13/60. If they had 180 pieces of candy, then the number of pieces that were fruit chews would be 180 × (13/60) = ***39 ***pieces.*

**♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.**

**Page 2 contains ONLY PROBLEMS. ♦**