Math-Essentials. Lesson 9-2: Counting Combinations

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1 Math-Essentials Lesson 9-2: Counting Combinations

2 Vocabulary Permutation: The number of ways a group of items can be arranged in order without reusing items.

3 Permutations What if you don t want to arrange all of the items? Instead you want to pick from a group of items but arrange only a portion of them. How many ways are there to do this? For example: Sean s band has 10 original songs. The recording company will only accept 6 songs on a demo CD. How many different ways can you choose 6 of the 10 and then arrange them on the demo disk? We call this a permutation of n items taken r at a time. 10P For Sean s CD: 10 permutate 6 n P r 6

4 Permutations. If we were making a permutation using the letters D, A, W, and G DAWG and WADG would be two distinct words. ORDER MATTERS!! (with permutations) a different order of members is a different group all together!! Permutation if arranging 4 items, pick any 4 items. Rearrange those 4 into a different order. If this new arrangement can be counted as a separate arrangement, it is a permutation.

5 I have 4 bills in my wallet: $1, $2, $5, $10 How many different sequences of bills can I take out of my wallet, if I only take 3 out? P ways 1, 2, 5 1, 5, 2 2, 1, 5 2, 5, 1 5, 1, 2 5, 2, 1 10, 2, 1 10, 1, 2 2, 10, 1 2, 1, 10 1, 10, 2 1, 2, 10 1, 10, 5 1, 5, 10 10, 1, 5 10, 5, 1 5, 1, 10 5, 10, 1 10, 2, 5 10, 5, 2 2, 10, 5 2, 5, 10 5, 10, 2 5, 2, 10 Each of these groups is just a permutation of the # of ways to arrange 3 different bills.

6 I have 4 bills in my wallet: $1, $2, $5, $10 How many different sums of money can I take out of my wallet, if I only take 3 bills out? 1, 2, 5 1, 5, 2 2, 1, 5 2, 5, 1 5, 1, 2 5, 2, 1 10, 2, 1 10, 1, 2 2, 10, 1 2, 1, 10 1, 10, 2 1, 2, 10 1, 10, 5 1, 5, 10 10, 1, 5 10, 5, 1 5, 1, 10 5, 10, 1 10, 2, 5 10, 5, 2 2, 10, 5 2, 5, 10 5, 10, 2 5, 2, 10 = $8 = $13 = $16 = $17 4 ways ORDER Doesn t MATTER!! a different order of pulling the same 3 bills out doesn t make a different sum. If order doesn t matter, then we have double counted the number of sums by the number of ways to arrange 3 different bills in order.

7 We call this new method of counting a combination. 1, 2, 5 1, 5, 2 2, 1, 5 2, 5, 1 5, 1, 2 5, 2, 1 10, 2, 1 10, 1, 2 2, 10, 1 2, 1, 10 1, 10, 2 1, 2, 10 P 3! n! Cr r! r!( n r)! n Pr n 1, 10, 5 1, 5, 10 10, 1, 5 10, 5, 1 5, 1, 10 5, 10, 1 10, 2, 5 10, 5, 2 2, 10, 5 2, 5, 10 5, 10, 2 5, 2, 10 = $8 = $13 = $16 = $ Using the multiplication principle of counting we must divide out the number of ways we have double counted.

8 Order Matters vs. Order Doesn t Matter Different order separate items n items taken r at a time Different Order not separate items must divide out the double counting n choose r items The symbol for this is: Permutation n! n P r ( n r)! n n C r P r n! r! r!( n r)! Combination n! r!( n r)!

9 Your turn: Permutation Combination You are tasked to count the number of ways the following items could occur. Decide if you will use a permutation or a combination (write P or C ) for each of the following: 3 people chosen out of a group of 10 to be the president, vice president and secretary of a club. 3 people chosen out of a group of 10 to members of a committee. The top 3 finishers of a race involving 20 runners. The 1 st, 2 nd, and 3 rd place finishers of a race involving 20 runners.

10 Key Question about Order Do I care if an item comes first or last (or somewhere in between) in the group of items I select? If you care about where/when the item is picked, order matters (use permutations) If not, order does not matter (use combinations)

11 Counting the # of ways to arrange choices: Order matters vs. Order Doesn t matter. ORDER MATTERS!! (with permutations) a different order of members is a different group all together!! Order matters: golf and flog are different words using the same 4 letter. Order doesn t matter: when summing a roll of two dice, getting a 3 first and a 5 second is the same as getting a 5 first and a 3 second. Since the order of rolling dice doesn t matter when finding the sum of the two dice we call this a combination.

12 Order Matters vs. Order Doesn t Matter Permutation Different order of the same items counted as a separate arrangement Different ways to line up people/things in order If you see the words in order in the question Different presidencies Different prizes based upon order of finish in a race

13 Order Matters vs. Order Doesn t Matter Combination Different order of the same items can not be counted as separate arrangement Different total scores Different total amounts of money Different hands of cards dealt in a game of cards (in games where you can rearrange the cards in your hand once they are dealt) Different committees of people

14 Combination: n C r n! r!( n r)! You are paying a for groceries at the store. You have the following bills: $100, $50, $20, $10, $5, $2, and $1. What are number of different sums of money that you can pull out of your if you pull out 3 bills without looking? 7 C 3 7! 3!(7 3)! 7! 3!(4)! 7*6*5*4! 35 3!(4)!

15 Combinations using your calculator n! 10! n C r 10C5 10 choose 5 r!( n r)! 5!(10 5)! Clear your screen Math button Scroll to PRB then enter 10 Select option 3 then hit 5 Now enter

16 C r n! Your turn: n r!( n r)! 7 choose 2 items =? 21 13C2? 78

17 Your turn: How many different committees with 5 members can be formed when choosing from 25 candidates? You are dealt 5 cards in a card game where you are allowed to rearrange the cards in your hand. How many different 5 card hands are possible? (you may rearrange the cards after they have been dealt). The number of ways 700 people can line up while in the lunch line.

18 Permutations or Combinations? Your turn: How many different ways can the 1 st, 2 nd, and 3 rd place trophies can be awarded to the top three contestants of 100 entrants. How many different 5 card hands can be dealt from a pack of 52 cards. (in this game you are not allowed to rearrange your cards in your hand after they have been dealt). You are shooting arrows at a target. Each ring on the target is worth a certain number of points. Your score is determined by the sum of points earned by shooting 3 arrows. How many different scores are possible using 3 arrows? (assume all 3 hit the target and you may hit the same ring more than once).

19 A Crash Course on Playing Cards for the Digital Age The group of cards that a player is given is called a hand 52 cards in a deck 2 colors: red ( ) and black ( ) 26 cards of each color 4 suits: Hearts, Diamonds, Spades, Clubs 13 cards in each suit 3 face cards: Jack, Queen, and King 10 numbered cards: 1 through 10 The 1 card is called an Ace 2 extra cards called the Jokers are sometimes added to the deck (making it a 54 card deck)

20 What did we learn? 1. The difference between discrete and continuous data. 2. The multiplication rule for counting ways things can be arranged in order. 3. The difference between a permutation and a combination when counting the ways to arrange things in order. 4. How to use a calculator to find the number of ways to arrange thing in order (permutation or combination).

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