Section 10.2 A Combinations. 2,598,960 How are combinations different from permutations? From a group of 5 students (Kyle, Joanne, Mary, Charlie, and.

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Presentation transcript:

Section 10.2 A Combinations

2,598,960

How are combinations different from permutations? From a group of 5 students (Kyle, Joanne, Mary, Charlie, and Vicki) we need to select 3… …who will get a first, second, and third place award for their work in math. …who will help their teacher to arrange the desks. Is order relevant to the arrangements?

nCrnCrnCrnCrCombinations of r objects, from a group of n items (order is NOT relevant to the problem)

nPrnPrnPrnPrPermutations of r objects, from a group of n items (order IS relevant to the problem)

How are combinations different from permutations? From a group of 5 students (Kyle, Joanne, Mary, Charlie, and Vicki) we need to select 3… …who will get a first, second, and third place award for their work in math. …who will help their teacher to arrange the desks. Is order relevant to the arrangements?

MULTIPLE EVENTS: when finding the number of ways both an event A and an event B can occur, you need to multiply. When finding the number of ways that event A or event B can occur, you add instead. a) 2 vegetarians and 1 meat… A restaurant serves omelets that can be ordered with any of the ingredients shown. a)Suppose you want to order exactly 2 vegetarian ingredients and 1 meat ingredient in your omelet. How many different types of omelets can you order?

MULTIPLE EVENTS: when finding the number of ways both an event A and an event B can occur, you need to multiply. When finding the number of ways that event A or event B can occur, you add instead. b) 2 vegetarian or 3 meat: A restaurant serves omelets that can be ordered with any of the ingredients shown. a)Suppose you want to order exactly 2 vegetarian ingredients and 1 meat ingredient in your omelet. How many different types of omelets can you order? b)Suppose you want to order 2 vegetarian ingredients or 3 meat ingredients in your omelet. How many different types of omelets can you order?

MULTIPLE EVENTS: when finding the number of ways both an event A and an event B can occur, you need to multiply. When finding the number of ways that event A or event B can occur, you add instead. c) 0 ingredients or 1 or 2 A restaurant serves omelets that can be ordered with any of the ingredients shown. a)Suppose you want to order exactly 2 vegetarian ingredients and 1 meat ingredient in your omelet. How many different types of omelets can you order? b)Suppose you want to order 2 vegetarian ingredients or 3 meat ingredients in your omelet. How many different types of omelets can you order? c)Suppose you want to order at most 2 ingredients in your omelet. How many different types of omelets can you order?

You are going on a vacation. You are packing your clothes and you can take as many as 7 t-shirts and 5 pairs of shorts. If you want to take exactly 3 pairs of shorts and 4 t-shirts, how many different ways can you pack?

Reword the problem: attend 9 games or 10 games or 11 games. Your school football team has 11 scheduled games for the season. You want to attend at least 9 games. How many different combinations of games can you attend?

Additional examples: Find the number of possible 5-card hands that contain the cards specified: a) 2 aces and 3 kings b) 5 hearts or 5 diamonds c) 3 hearts (hint: this means 3 hearts and 2 “non-hearts”) d) 4 face cards (hint: this means 4 face cards and 1 “non-face” card)