10.2.notebook. February 24, A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit.

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1 Section 10.2 It is not always important to count all of the different orders that a group of objects can be arranged. A combination is a selection of r objects from a group of n objects where the order is not important. Combinations 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 given by the formula: Example 1 A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit. If the order in which the cards are dealt is not important, how many different 5 card hands are possible? 1

2 Example 2 Coach Evans has a gym class of 9 students that are wanting to play some wiffleball and he needs to put them into teams. 4 of the students are male and the other 5 are female. If the order in which they chosen doesn't matter, how many different teams of 3 students can he choose? 2

3 Multiple Events 1. When finding the number of ways both an event A and an event B can occur, you need to multiply. 2. When finding the number of ways that event A or event B can occur, you add Example 3 The Student Senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshmen. a. How many different committees of exactly 2 seniors and 2 juniors can be chosen? b. How many different committees of four seniors or 2 freshmen can be chosen? Try: The local movie rent al store is having a special on new releases. The new releases consist of 12 comedies, 8 action, 7 drama, 5 suspense, and 9 family movies. a. You want exactly 2 comedies and 3 family movies. How many different movie combinations can you rent? b. You want 3 suspense movies or 5 action movies. How much movie combinations can you rent? 3

4 More Examples 4. A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit. a. If the order is not important in which the cards are dealt, how many different 7 card hands are possible? b. In how many 7 card hands are all 7 cards the same color? 4

5 Try There are fourteen juniors and twenty three seniors in the Service Club. The club is to send four representatives to the State Conference. a.) How many different ways are there to select a group of four students to attend the conference? b.) If the members of the club decide to send two juniors and two seniors, how many different groupings are possible? There are 12 boys and 14 girls in Mrs. Tabb's math class. She is looking to select a team of students from the class to work on a group project. a. The team is to consist of 1 girl and 2 boys. b. The team is to consist of 3 boys or 5 girls c. The team is to consist of 5 students Using the standard deck of playing cards, how many different 5 card hands are possible if you want: a. 3 jacks and 2 other cards b. 5 clubs or 5 spades 5

6 Many of the relationships among combinations can be seen in the array of numbers known as Pascal s Triangle. n =0 (0th row ) 1 n = 1 (1st row ) 1 1 n= 2 (2n d row ) n = 3 (3rd row ) n = 4 (4th row ) n = 5 (5th row) n = 6 (6th row) Example 6 From a collection of 7 St. Louis Cardinals baseball caps, you want to trade 3. Use Pascal s triangle to find the number of combinations of 3 caps that can be traded. Try: Out of 5 finalists, your class must choose 3 class representatives for an upcoming 3 on 3 basketball class tournament. Use Pascal s triangle to find the number of combinations of 3 students that can be chosen as representatives. 6

7 Binomial Expansions We ll now explore the connection between Pascal s triangle and binomial expansions. (Binomial Theorem) The power of the binomial refers to the row of Pascal's Triangle and this will tell you your coefficients Start with the first term, raise it to the n power the power and decrease by 1 as you move left to right Take the second term, raise it to the zero power, and increase by 1til you reach the n power as you move left to right Binomial sum has all plus signs Binomial difference has alternate signs starting with a minus Example 7 Expand (x + y)5 Example 8 Expand (3x+2)3 Example 9 Expand (4 2y)3 7

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