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10.2 Combinations and Binomial Theorem What you should learn: Goal1 Goal2 Use Combinations to count the number of ways an event can happen. Use the Binomial Theorem to expand a binomial that is raised to a power. 10.2 Combinatins and Binomial Theorem

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In the last section we learned counting problems where order was important For other counting problems where order is NOT important like cards, (the order youre dealt is not important, after you get them, reordering them doesnt change your hand) These unordered groupings are called Combinations 12.2 Combinatins and Binomial Theorem

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A Combination is a selection of r objects from a group of n objects where order is not important 12.2 Combinatins and Binomial Theorem

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Combination of n objects taken r at a time The number of combinations of r objects taken from a group of n distinct objects is denoted by n C r and is: 12.2 Combinatins and Binomial Theorem

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For instance, the number of combinations of 2 objects taken from a group of 5 objects is 2 12.2 Combinatins and Binomial Theorem

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Finding Combinations In a standard deck of 52 cards there are 4 suits with 13 of each suit. If the order isnt important how many different 5-card hands are possible? The number of ways to draw 5 cards from 52 is = 2,598,960

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In how many of these hands are all 5 cards the same suit? You need to choose 1 of the 4 suits and then 5 of the 13 cards in the suit. The number of possible hands are: 12.2 Combinatins and Binomial Theorem

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How many 7 card hands are possible? How many of these hands have all 7 cards the same suit? 12.2 Combinatins and Binomial Theorem

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When finding the number of ways both an event A and an event B can occur, you multiply. When finding the number of ways that an event A OR B can occur, you +. 12.2 Combinatins and Binomial Theorem

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Deciding to ADD or MULTIPLY A restaurant serves omelets. They offer 6 vegetarian ingredients and 4 meat ingredients. You want exactly 2 veg. ingredients and 1 meat. How many kinds of omelets can you order? 12.2 Combinatins and Binomial Theorem

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Suppose you can afford at most 3 ingredients How many different types can you order? You can order an omelet with 0, or 1, or 2, or 3 items and there are 10 items to choose from. 12.2 Combinatins and Binomial Theorem

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Counting problems that involve at least or at most sometimes are easier to solve by subtracting possibilities you dont want from the total number of possibilities. 12.2 Combinatins and Binomial Theorem

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Subtracting instead of adding: A theatre is having 12 plays. You want to attend at least 3. How many combinations of plays can you attend? You want to attend 3 or 4 or 5 or … or 12. From this section you would solve the problem using: Or……

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For each play you can attend you can go or not go. So, like section 10.1 it would be 2*2*2*2*2*2*2*2*2*2*2*2 =2 12 And you will not attend 0, or 1, or 2. So:

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0 C 0 1 C 0 1 C 1 2 C 0 2 C 1 2 C 2 3 C 0 3 C 1 3 C 2 3 C 3 4 C 0 4 C 1 4 C 2 4 C 3 4 C 4 Etc… 12.2 Combinatins and Binomial Theorem

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Pascal's Triangle! 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Etc… This describes the coefficients in the expansion of the binomial (a+b) n

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(a+b) 2 = a 2 + 2ab + b 2 (1 2 1) (a+b) 3 = a 3 (b 0 )+3a 2 b 1 +3a 1 b 2 +b 3 (a 0 ) (1 3 3 1) (a+b) 4 = a 4 +4a 3 b+6a 2 b 2 +4ab 3 +b 4 (1 4 6 4 1) In general… 12.2 Combinatins and Binomial Theorem

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(a+b) n (n is a positive integer)= n C 0 a n b 0 + n C 1 a n-1 b 1 + n C 2 a n-2 b 2 + …+ n C n a 0 b n = 12.2 Combinatins and Binomial Theorem

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(a+3) 5 = 5 C 0 a 5 3 0 + 5 C 1 a 4 3 1 + 5 C 2 a 3 3 2 + 5 C 3 a 2 3 3 + 5 C 4 a 1 3 4 + 5 C 5 a 0 3 5 = 1a 5 + 15a 4 + 90a 3 + 270a 2 + 405a + 243 12.2 Combinatins and Binomial Theorem

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Assignment 12.2 Combinatins and Binomial Theorem

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