MATH 5007 - Mathematical and Statistical Sciences



Donuts!!

Feeling hungry, you head out to your local donut shop to pick up a dozen and (since you are now enraptured by counting) you start to wonder....how many different dozens could I order?

As you count, the only rule is that the order of donuts in the box does not matter, only the number of donuts of each type.

1. Suppose the shop offers only 1 type of donut, how many different dozens can be made?

2. Suppose the shop offers 2 different types of donuts, chocolate and glazed. How many different dozens can be made? (Remember: CCCCCCGGGGGG is the same as CGCGCGCGCGCG).

3. Suppose that there are 3 options: Chocolate, Glazed and Pickled.

a) How many dozens can you make with zero chocolate donuts?

b) How many dozens can you make with one chocolate donut?

c) How many dozens can you make with eight chocolate donuts?

d) Using the same reasoning as in (a) – (c), determine the total number of possible dozens.

According to they sell 53 different types of donuts. It may take a while to count all of the possibilities.

Idea: Suppose you want to put some effort into organizing your donut box. Here are some rules:

a) Put all of the donuts of the same flavor together.

b) Decide in advance the order you will put the flavors in the box. For our three-donut shop, I will always put chocolate, then glazed, then pickled.

c) Place a divider between each of the flavors (even if you have no donuts of a certain flavor. For instance, a dozen with 3 chocolate, 5 glazed and 4 pickled would be represented by

CCC | GGGGG | PPPP

and a dozen with 6 chocolate, 0 glazed and 6 pickled would be represented by

CCCCCC | | PPPPPP.

4. Suppose now you are in a shop with four varieties: Chocolate (C), Glazed (G), Pickled (P) and Blueberry (B). Write out each of the following dozens, including dividers, in the manner described above. Assume the order of the flavors in each box is C, G, P, B.

a) 3 of each flavor.

b) 2 chocolate, 5 glazed, 3 pickled, 2 blueberry.

c) 7 chocolate, 4 pickled, 1 blueberry.

d) 12 pickled.

I can write my boxes out in an even simpler way. Once I know the order of my flavors, I don't even need to use initials to distinguish between the types. For instance, if I know that my flavor order is C, G, P, B then the box with three of each flavor can be written as

x x x | x x x | x x x | x x x

and the box with 2 chocolate, 5 glazed, 3 pickled and 2 blueberry can be written as

x x | x x x x x | x x x | x x.

5. Suppose you are still in a shop with four varieties: Chocolate (C), Glazed (G), Pickled (P) and Blueberry (B). Write out each of the following dozens, using 'x' to represent a donut and '|' to represent a divider. Assume the order of the flavors in each box is C, G, P, B.

a) 7 chocolate, 4 pickled, 1 blueberry.

b) 12 pickled.

c) 6 chocolate, 6 blueberry.

6. We are now ready to count the number of possible dozens in our shop with four flavors. Think of representing the possible dozens using our 'x' and '|' method.

a) How many dividers will you need to separate the four flavors?

b) If you think of your box as having one space for each donut and one space for each divider, how many spaces does the box have in total?

c) How many different ways are there to place the number of dividers from (a) into the number of spots from (b)?

d) Explain how your answer from (c) counts the number of possible dozens of donuts.

7. Suppose you are in a donut shop with n different flavors. Think of representing the possible dozens using our 'x' and '|' method.

a) How many dividers would you need to separate the n flavors”

b) If you think of your box as having one space for each donut and one space for each divider, how many spaces does the box have in total?

c) How many different ways are there to place the number of dividers from (a) into the number of spots from (b)?

d) Explain how your answer from (c) counts the number of possible dozens of donuts.

8. How many possible dozens are there at Dunkin Donuts (53 different flavors)?

a) How many dividers?

b) How many donuts?

c) How many total spots in the box?

Total number of dozens:

Practice Problems

Donuts!

1. You go to a Dunkin Donuts which offers 5 different type of donuts,

a) How many different dozens are possible?

b) How many different dozens have exactly 4 glazed?

c) How many different dozens have at least 4 glazed?

d) How many different dozens have at least one of each type?

e) How many different dozens have at least two of each type?

f) Suppose that, instead of 12 donuts, you wanted to buy 20. With no other restrictions, how many ways could you do this?

2. You have 20 candies to distribute to 5 children,

a) How many different ways can you distribute the candies?

b) How many different ways can you distribute the candy so that the first child gets exactly 8 candies?

c) How many different ways can you distribute the candy so that the first child gets at least 8 candies?

d) How many different ways can you distribute the candy so that each child gets at least 1 candy?

e) How many different ways can you distribute the candy so that each child gets at least 2 candies?

3. Given the equation

x1 + x2 + x3 + x4 + x5 = 20

a) How many different non-negative integer solutions are possible?

b) How many have x1 = 8?

c) How many have x1 ≥ 8?

d) How many have each xi ≥ 1?

e) How many have xi ≥ 2 for each?

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download