Math 105 Week 5 Notes
Math 105 Week 5 Notes MATH 105-02
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This 5 page Class Notes was uploaded by Danielle Kelly on Friday February 13, 2015. The Class Notes belongs to MATH 105-02 at Washington State University taught by Spencer Payton in Spring2015. Since its upload, it has received 268 views. For similar materials see Exploring Mathematics in Mathematics (M) at Washington State University.
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Date Created: 02/13/15
Math 105 February 9 2015 Permutations Combinations Of ways to select ritems out of n iteml Repetition is NOT allowed Order is important Order is not important Arrangement of n Subsets of n items taken Items rat a time rat a time NpR nnr nr NPRr nrnr Clue Words order Clue Words Set group Arrangements schedule sample select Note If repetition is allowed just use the Fundamental Counting Principle Example In how many ways can a mother distribute 3 different toys among her seven children if a child may receive anywhere from none to all 3 toys use the Fundamental Counting Principle 7 7 lsttoy 2ndt0y 7 73 343 ways the 3 toys could be distributed to the 7 kids 3 rd toy For this problem the rst toy has 7 children it could be given to the 2nCI toy has 7 children it can be given to that is because any child could get anywhere from all to none of the toys So the third two also has 7 different children it can be given to 7 x 7 x 7 73 343 Example How many different 3 member committees could club N appoint so that exactly one woman is on the committee Club N Alan Bill Cathy David and Evelyn Step 1 choose 1 woman Step 2 choose 2 men Using the fundamental counting principle 2 8 2C1 111 2 3 b 3C2 211 3X2X1 2X1 3 2 3 M 6 different possible committee 2men Example 10 members of the Alpha Beta Gamma fraternity will be selected to attend a special even this weekend How many ways can these 10 members be selected if there are a total of 48 members 0 10 distinct men 0 order doesn39t matter 48 members and you want to pick 10 48P10 48 48P10 10138 6540715896 different ways to pick the 10 members When the 10 fraternity men arrive at the event 4 of them are selected to escort the homecoming queen candidates How many ways can this selection be made 0 4 distinct men 0 Order does matter because different orders will pair men with different women 0 This question is an implied assumption 10 members and you want to pick 4 10P4 10 10P4 4 5040 possible ways to pick the 4 men Section 105 Problems including quotnotquot and quotorquot A U nA nU nA39 The number in A the numbers in U the numbers in the compliment of A AUA39U andA n A39 Example For S a b c d e f Find the number of proper subsets 2x2x2x2x2x2 26 64 subsets A subset of this set that is not a proper subset it the set itself so that means there are 63 proper subsets for this set Math 105 Februarv 11 2015 105 Problems lnvolvino quotNotquot and quotorquot Example If four coins are tossed in how many ways can at least one tail be obtained There is only one set that doesn39t have mtails and that set is HHHH If there is not at least one tail than there are no tails at all 2 2 2 2 4 I lsttoss 2ndtoss 3rdtoss 4th toss 2222 2 16 p05539ble OUt comes Out of these 16 possible outcomes one of them is a set of all heads So that means you subtract 1 from 16 which leaves you with 15 There are 15 possible outcomes that will have at least one tail Example Four friends are boarding a plane There are only ten seats left three of which are aisle seats How many ways can the four people arrange themselves in available seats so that at least one of them sits on the aisle 0 The word arrange means that order matters which means you use permutations quotat least one aisle seatquot means the opposite of quot no aisle seatquot I t A 1 way no one sits in the aisle U A Total number of ways to arrange 4 people among 10 seats 10 10987654321 ways Total number of ways to arrange 4 people among the nonaisle seats There are 7 nonaisle seats 7 7654321 7P4 47 4321 756 840 ways nAnU nA 5040 840 4200 arrangements with at least 1 person sitting in an aisle seat Additive principle of counting nAUB nAnBnA n B lfA and B are disjoint A n Bo then nAUBnA nB Example How many vecard poker hands consist of all clubs or all red cards Clubs 0 Red cards 93 nall cubs or all red cards nall clubs nred cards 13 clubs in a deck 26 red cards in the deck 13C5 26C5 128765780 67067 possible ways of having either all clubs or all red cards Example Congressional members Men M Women W Total Republican R 5 3 8 Democrat D 4 6 10 Total 9 9 18 If one of these members is chosen randomly to be spokesperson for the group in how many ways could that person be a Democrat or a women nD or W nD nw nD and W There are 10 total democrats There are 9 total women There are a total of 6 women who are also democrats 10 democrats 9 women 6 democrat women 13 ways to pick a democrat g a women Example How many threedigit counting numbers are multiples of 2 or multiples of 5 0 Multiple of 2 must end in a 02468 Multiple of 5 must end in a 0 or 5 0 Multiple of 2 and 5 must end in 0 Multiples of 2 9105 450 3digit multiples of 2 Multiples of 5 9102 180 3digit multiples of 5 Multiples of 2 and 5 9101 90 3digit multiples of 2 and 5 Example A single card is drawn from a standard 52card deck A How many ways could the card be a heart or a king There are 13 hearts in the deck 4 hearts in the deck and one king of hearts in the deck 13 hearts 4 kings 1king of hearts 16 possible ways to draw a heart or a king B How many ways could it be a club or face card There are 13 clubs in a deck 12 face cards in a deck and 3 club face cards in a deck 13clubs 12 face cards 3 club face cards 22 possible ways to draw a club or face card
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