Explanation Definition: Fundamental counting principle: Suppose one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is formulated as m ⋅ n . It can be extended to three or more events. Permutation: A permutation is an arranged in the definite order of a number of objects will be taken in some or all at a time. The number of permutation of n different items takes from r items. The formula is, n P r = n ! ( n − r ) ! . Where, 0 < r ≤ n . Combination: The number of possible combination of r objects from a set of n objects. The formula is, n C r = n ! r ! ( n − r ) ! . Given: Customer: So what’s this new deal? Pizza chef: Two pizzas Customer [Toward four-year-old boy]: Two pizzas. Write that down. Pizza chef: And on the two pizza choose any toppings-up to five [ from a list of 11 toppings ]. Older boy: Do you… Pizza chef: …have to pick the same toppings on each pizza? No! Four-year-old-math whiz : Then the possibilities are endless. Customer: What do you mean? Five plus five are ten. Math whiz: Actually, there are 1, 048, 576 possibilities. Customer: Ten was just a ballpark figure. Old man: You go that right. Calculation: The number of possibility for pizza without any double toppings: The person has to order a single pizza with 5 toppings out of 11. The boy can order a pizza with 0 toppings, 1 toppings, 2 toppings, 3 toppings, 4 toppings or 5 toppings. It can be calculate by using the combinations. Compute the number of possibility for pizza without any double toppings. 11 C 0 + 11 C 1 + 11 C 2 + 11 C 3 + 11 C 4 + 11 C 5 = 1 + ( 11 ! 1 ! ( 11 − 1 ) ! ) + ( 11 ! 2 ! ( 11 − 2 ) ! ) + ( 11 ! 3 ! ( 11 − 3 ) ! ) + ( 11 ! 4 ! ( 11 − 4 ) ! ) + ( 11 ! 5 ! ( 11 − 5 ) ! ) = 1 + ( 11 ! 1 ! ( 10 ) ! ) + ( 11 ! 2 ! ( 9 ) ! ) + ( 11 ! 3 ! ( 8 ) ! ) + ( 11 ! 4 ! ( 7 ) ! ) + ( 11 ! 5 ! ( 6 ) ! ) = 1 + 11 + ( ( 10 ) ( 11 ) 2 ) + ( ( 9 ) ( 10 ) ( 11 ) 6 ) + ( ( 8 ) ( 9 ) ( 10 ) ( 11 ) 24 ) + ( ( 7 ) ( 8 ) ( 9 ) ( 10 ) ( 11 ) 120 ) = 1 + 11 + 55 + 165 + 330 + 462 = 1024 Thus, the single pizza can be ordered by 1024 ways. The number of ways to order a pizza with two double toppings: There are 11 toppings are available. Need to choose 4 toppings to pizza for two double toppings. The number of ways is 11 C 2 . Thus, there are 9 possible remaining toppings available. Compute the number of ways to order a pizza with two double toppings

Nature of Mathematics (MindTap Course List)

13th Edition

ISBN: 9781133947257

Author: karl J. smith

Publisher: Cengage Learning

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Counting Principles

In statistics, the counting principle can be said to be the concept in which the quantity of an item is determined. There can be a single item, or a group of items, or items that are related to each other. In every possible set of items, the quantity of …

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Chapter 12.3, Problem 58PS

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The number of possible different pizza with two double toppings.

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Chapter 12.3, Problem 57PS

Chapter 12.3, Problem 59PS

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Nature of Mathematics (MindTap Course List)

Ch. 12.1 - IN YOUR OWN WORDS What is a permutation? What is...Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Evaluate each expression in Problems 3-32. 9P2 Ch. 12.1 - Evaluate each expression in Problems 3-32. 9P3 Ch. 12.1 - Evaluate each expression in Problems 3-32. 9P4 Ch. 12.1 - Prob. 8PS Ch. 12.1 - Evaluate each expression in Problems 3-32. 52P3 Ch. 12.1 - Evaluate each expression in Problems 3-32. 4P4

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The number of possible different pizza with two double toppings.

Solution Summary: The author explains the basic counting principle: if one event can occur in m ways, the other event cannot occur. Permutation is an arranged in the definite order of a number of objects.

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To find: The number of possible different pizza with two double toppings.

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To find:

The number of possible different pizza with two double toppings.