Questions, Answers, and Discussion

## Look for patterns and answer the following questions:

How many different 1-topping pizzas can you order when choosing from

among 8 toppings?You can order 8 different 1-topping pizzas:

You can find the answer by listing the 8 possible pizzas, as shown above, or think: how many different pizza combinations can I make by choosing 1 topping from a set of 8 toppings?

Using Pascal's triangle, find place 1 in row 8: 8 ways. [Remember that the first number (1) in each row is place 0.]

Because of this choosing property, the binomial coefficient [8:1] is usually read "eight choose one."

How many different 7-topping pizza combinations can you order from

a set of 8 toppings?You can order 8 different 7-topping pizzas:

1. ACGMOPS

anchovies, extra cheese, green peppers, mushrooms, olives, pepperoni, sausage2. ACPMOPT

anchovies, extra cheese, green peppers, mushrooms, olives, pepperoni, tomatoes3. ACPMOST

anchovies, extra cheese, green peppers, mushrooms, olives, sausage, tomatoes4. ACGMPST

anchovies, extra cheese, green peppers, mushrooms, pepperoni, sausage, tomatoes5. ACGOPST

anchovies, extra cheese, green peppers, olives, pepperoni, sausage, tomatoes6. ACMOPST

anchovies, extra cheese, mushrooms, olives, pepperoni, sausage, tomatoes7. AGMOPST

anchovies, green peppers, mushrooms, olives, pepperoni, sausage, tomatoes8. CGMOPST

extra cheese, green peppers, mushrooms, olives, pepperoni, sausage, tomatoes

You can find this answer by listing the 8 possible pizzas, as shown above, or think: how many different 7-topping pizza combinations can I make from a set of 8 toppings?

Using Pascal's triangle, find place 7 in row 8: 8 ways.

Because of this choosing property, the binomial coefficient [8:7] is usually read "eight choose seven."

How is the total possible number of 1-topping pizzas related to

the total possible number of 7-topping pizzas? Why?

When you order a 1-topping pizza, you choose not to use 7 toppings.

When you order a 7-topping pizza, you choose not to use 1 topping.

The number of total possible choices is the same in each case: 8.

How many different pizza combinations can you make using 2 toppings?

You can order 28 different pizza combinations when you choose 2 toppings from a set of 8 toppings:

A = anchovies, C = extra cheese, G = green peppers, M = mushrooms,

O = olives, P = pepperoni, S = sausage, T = tomatoes

AC AG AM AO AP AS AT CG CM CO CP CS CT GM GO GP GS GT MO MP MS MT OP OS OT PS PT ST You can find this answer by listing the 28 possible pizzas as shown, or think: how many different pizza combinations can I order if I choose 2 toppings from a set of 8 toppings?

Using Pascal's triangle, find place 2 in row 8: 28 ways.

Because of this choosing property, the binomial coefficient [8:2] is usually read "eight choose two."

How many different pizzas can you make using 6 toppings?

You can order 28 different pizza combinations when you choose 6 toppings from a set of 8 toppings:

A = anchovies, C = extra cheese, G = green peppers, M = mushrooms,

O = olives, P = pepperoni, S = sausage, T = tomatoes

GMOPST CMOPST CGOPST CGMPST CGMOST CGMOPT CGMOPS AMOPST AGOPST AGMPST AGMOST AGMOPT AGMOPS ACOPST ACMPST ACMOST ACMOPT ACMOPS ACGPST ACGOST ACGOPT ACGOPS ACGMST ACGMPT ACGMPS ACGMOT ACGMOS ACGMOP You can find this answer by listing the 28 possible pizzas, as shown, or think: how many different pizza combinations can I order if I choose 6 toppings from a set of 8 toppings?

Using Pascal's triangle, find place 6 in row 8: 28 ways.

Because of this choosing property, the binomial coefficient [8:6] is usually read "eight choose six."

How is the total possible number of 2-topping pizzas related to

the total possible number of 6-topping pizzas? Why?When you order a 2-topping pizza, you choose not to use 6 toppings.

When you order a 6-topping pizza, you choose not to use 2 toppings.

The number of possible choices is the same in each case: 28.

Can you find these numbers in Pascal's triangle? Look at row 8:

Can you use Pascal's triangle to help you find the number of different

pizza combinations you can order with three, four, or five toppings?

To find the number of different 3-topping pizza combinations you can order, think: How many different combinations of 3 toppings can I make from a set of 8 toppings? Using Pascal's triangle, find place 3 in row 8: 56 combinations. To find the number of different 4-topping pizza combinations you can order, think: How many different combinations of 4 toppings can I make from a set of 8 toppings? Using Pascal's triangle, find place 4 in row 8: 70 combinations. To find the number of different 5-topping pizza combinations you can order, think: How many different combinations of 5 toppings can I make from a set of 8 toppings? Using Pascal's triangle, find place 5 in row 8: 56 combinations. What's the total number of different pizza combinations that can be made

given a choice of 8 toppings?

1 pizza with no toppings

8 different pizza combinations with 1 topping

28 different pizza combinations with 2 toppings

56 different pizza combinations with 3 toppings

70 different pizza combinations with 4 toppings

56 different pizza combinations with 5 toppings

28 different pizza combinations with 6 toppings

8 different pizza combinations with 7 toppings

1 pizza with eight toppings

Use Pascal's triangle to find the sum of the numbers in row 8:

1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256

Now let's try a different approach to the problem. Antonio could have

helped the Pascalini's if he had asked the following questions:

Do you want anchovies?

Do you want extra cheese?

Do you want green peppers?

Do you want mushrooms?

Do you want olives?

Do you want pepperoni?

Do you want sausage?

Do you want tomatoes?

How could this information help you to find the total number of different pizza combinations that can be ordered?

There are two possible answers to each of the 8 questions, yes or no. We can express the total possible ways to answer these 8 questions as:

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 ^{8}= 256

Notice that the sum of the entries in the 8th row of Pascal's triangle can also be expressed as

2 ^{8}= 256

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