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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."

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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, sausage

2. ACPMOPT               anchovies, extra cheese, green peppers, mushrooms, olives, pepperoni, tomatoes

3. ACPMOST               anchovies, extra cheese, green peppers, mushrooms, olives, sausage, tomatoes

4. ACGMPST               anchovies, extra cheese, green peppers, mushrooms, pepperoni, sausage, tomatoes

5. ACGOPST               anchovies, extra cheese, green peppers, olives, pepperoni, sausage, tomatoes

6. ACMOPST               anchovies, extra cheese, mushrooms, olives, pepperoni, sausage, tomatoes

7. AGMOPST               anchovies, green peppers, mushrooms, olives, pepperoni, sausage, tomatoes

8. 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.

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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."

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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."

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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:

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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

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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 = 2⁸ = 256

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

28 = 256

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