Combinatorics Scribe

Combinatorics PreTest Are you ready?

Question 1: Seven teams compete in a men's hockey league. If each team plays each other twice , how many games are necessary to complete the league schedule? (a) 42 (b) 21 (c) 84 (d) 52 quot;Why...?quot;

Solution 1 : Using the 'choose' formula There are 7 teams and Multiply by 2 because you have to choose 2 to they have to play each play against each other other twice ( C ) 2 • 2 7 (21) • 2 42 is your final answer, 42 you go back to the original question and circle the answer 42

Solution 2: Making a table 1234567 When you add up all the J's and O', XJ J J J J J 1 The blue J's you get 42. represent the OX J J J J J 2 Therefore, there second round of OOX J J J J 3 are 42 games to games they play complete the O OOX J J J 4 league schedule OOOOX J J 5 O OOO O X J 6 7 O O O O OO X The red x's are there The green O's represent because the team's can't one round of games play themselves, so they play those slots don't count

Question 2: The sum of the seventh row of Pascal's triangle is the same solution to: (a) The number of solutions a student could get if they guessed on six questions of a multiple choice exam, each question having four answers. (b) The number of solutions a student could get if they guessed on six questions of a true and false exam. (c) The number of solutions a student could get if they guessed on seven questions of a multiple choice exam, each question having four answers. (d) The number of solutions a student could get if they quessed on seven questions of a true and false exam. quot;How do you know...?quot;

Solution Ask yourself, 'which one of these solutions will equal 64?' Sixth row Let's try (b)! sum = 64 There are 6 questions, so you have to do this 6 times Pascal's Triangle =O! Look at that... The 6 ( C ) 1 = 64 answer is 64! Now go and 2 circle answer (b) Since it's a true and false Out of the two choices, exam, there's only two choose one answers to choose from

Question 3: Eight friends go to dinner. How many ways can they be seated at a circular table if three of them insist on sitting together? ANSWER: 720 quot;How did you do that?!quot;

Solution There are eight friends and three of The formula for circular them love each other oh so much. permutations is (n1)! Let's put them in a bag! In the bag there are There are 6 three things that things to arrange can be arranged ( 6 1)! 3! Table (5)! 3! 720

Question 4: A multiple choice exam has 20 questions, each with four possible answers, and 10 additional questions, each with five possible answers. How many different answer sheets are possible? ANSWER: CRAZY BIG NUMBER quot;How come..?quot;

Solution There's 20 questions, so you have to do this The first 20 give There's 10 questions, 20 times you 4 choices, so you have to do choose 1 of them this 10 times 10 20 ( C ) 1 ( C ) 1 • 5 4 The next 10 questions have 5 choices, choose 1 of them This gives you a 'CRAZY BIG NUMBER'. I'm not really sure how to write it... but it's a 'CRAZY BIG NUMBER'!

Question 5: In a 52 card deck, how many 5 card poker hands are possible that have exactly two pair? one pair? ANSWER: Two pair: 123552 One pair: 1098240 quot;Are you serious?!quot;

Solution: Two Pair quot;I'm serious.quot; Suits for the A different face value first pair for the last card ( C ) 2 ( C ) ( C ) 2 ( C ) 1 ( C ) 1 • • • • 4 13 4 11 2 4 2 face values A suit for the Suits for the (one for each pair) last card second pair 123552 different possibilities

Solution: One Pair * 3 cards left, so 3 different 1 face value you do it 3 times* face values for the pair 3 ( C ) • ( C ) 2 • ( C ) ( C ) 1 • 4 4 13 12 1 1 1 suit for the three 2 suits for the pair remaining cards 1098240 different possibilities

... And that's the end of this mindblowing unit. Unit test is tomorrow! Study hard, eat some Lucky Char m s, and bring a lucky rabbit's foot, but not a real rabbit's foot because... THAT'S CRUEL! (Enjoy! =D )

Next Scribe is... Paulo Bear! Insert Polar Bear Image Here

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Combinatorics Scribe

by jasminprecal40s on Dec 06, 2007

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