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Exploring Pascal || Student Center || Teachers' Place
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Exploring Pascal's Triangle

Antonio's Pizza Palace

Intermediate or Advanced Level

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, 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.


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 = 28 = 256

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

  28 = 256 
 
Questions? Write to the workshop facilitators.

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Math Forum * * * * 27 April 1998