STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 1 2 Multiplication Principle Fundamental Rule of Counting If there are n events and event i can occur in N i possible ways then the number of ways in which the sequence of n events may occur is N 1 N 2 N n 1 Manager of a radio station decided that every day the broadcast will start with one of the 9 Beethoven Symphonies followed by one of Mozart s 27 Piano Concertos followed by one of Schubert s 15 String Quartets Approximately how many years can the station do that without repeating the program 9 Beethoven Symphony 27 Mozart s Piano Concerto 15 3645 days Schubert s String Quartet 3645 days 10 years 1 The call letters of radio and television stations in the United States begin with either K or W Those west of the Mississippi River start with K and those east of it with W a Some stations such as KID in Idaho Falls Idaho and WOW in Omaha Nebraska have 3 call letters How many sets of call letters having 3 letters re possible 2 26 26 1 352 b Most stations that were licensed after 1927 have 4 call letters such as KUZZ in Bakersfield California and WXYZ in Detroit Michigan How many sets of call letters having 4 letters are possible 2 26 26 26 35 152 c How many sets of call letters having 4 letters are possible if we are not allowed to repeat letters 2 25 24 23 27 600 2 In how many orders can the names of 5 candidates for the same office be listed on a ballot 5 4 3 2 1 1st 2nd 3rd 4th 120 5th n 1 2 n 1 n n n n 1 2 1 0 1 n is the number of ways to rearrange reorder n distinct items For example there are 7 5 040 different ways to arrange 7 books on a bookshelf 3 How many ways are there of scrambling the letters of the word SCRAMBLE There are 8 letters in the word SCRAMBLE none of them repeating Therefore there are 8 40 320 different ways to rearrange the letters 4 Eight horses are entered in a race in which bets are placed on which horse will win place and show that is finish first second and third Suppose that the race is run and there are no ties a In how many orders can all eight horses finish the race 8 40 320 different ways for eight horses to finish the race b In how many ways can the win place and show be taken 8 7 6 336 1st 2nd 3rd 8 7 6 8 7 6 5 4 3 2 1 5 4 3 2 1 8 5 Permutations are the possible ordered selections of r objects out of a total of n objects The number of permutations of n objects taken r at a time is nPr n n r For example there are 11 P 4 11 10 9 8 7920 different ways to appoint a president a vice president a secretary and a treasurer for a club that has 11 members 5 At Momma Leona s Pizza you can get a pizza with or without each of eight different toppings How many different three topping pizzas can you get at Momma Leona s Pizza if each topping can be put on a pizza at most once The order in which the toppings toppings are selected does not matter There are 8 P 3 336 ordered selections of three toppings out of eight Each unordered selection is repeated 6 3 times ABC ACB BAC BCA CAB CBA Therefore the number of unordered selections of three toppings out of eight is 8 336 8 P3 56 3 3 8 3 6 Combinations are the possible selections of r items from a group of n items regardless of the order of selection The number of combinations of n objects taken r at a time is n n r r n r C n r Pascal Triangle 1 1 1 1 1 2 1 1 1 1 1 9 10 For example 45 210 8 C 1 8 8 C 5 56 1 8 84 210 8 C 0 1 7 28 126 252 1 21 56 126 1 6 35 70 84 120 5 15 35 56 36 1 10 20 21 28 4 10 15 7 1 6 5 6 8 3 4 1 1 3 1 36 120 9 45 8 C 2 28 8 C 6 28 1 1 10 1 8 C 3 56 8 C 7 8 8 C 4 70 8 C 8 1 0 1 2 3 4 5 6 7 8 9 10 n n r n r x y n n n n n 1 2 2 n 2 n n n 1 n 1 n n 1 0 n n n n 2 n k k 0 n k n k x y k k 0 n n k 0 1 k k 0 order of the selection is important order of the selection is not important repetitions allowed w replacement nr n r 1 Cr repetitions not allowed w o replacement n n P r n r n Cr n r n r 6 The Baskin Robbins Ice Cream Stores have 31 flavors of ice cream a How many different 3 scoop ice cream cones are possible if you are allowed to repeat flavors and want the scoops put on the cone in a particular order w replacement 31 3 29 791 order of the selection is important OR 31 1st b 31 31 29 791 2nd 3rd How many different 3 scoop ice cream cones are possible if each scoop is a different flavor and you want the scoops put on the cone in a particular order w o replacement order of the selection is important 31 31 P 3 28 26 970 OR 31 1st c 30 29 26 970 2nd 3rd How many different 3 scoop cones are possible if each scoop is a different flavor and you don t care about their order on the cone w o replacement order of the selection is not important 31 31 31 C 3 3 3 28 4 495 d How many different 3 scoop ice cream cones are possible if you are allowed to repeat flavors but the order in which the scoops are placed into the cone is not important order of the selection is not important w replacement 33 31 3 1 C 3 3 33 3 30 5 456 7 To play Michigan Lotto a person must pick 6 numbers from 49 numbers a If the player matches all 6 numbers 6 of 6 drawn he she wins the grand prize jackpot Find the probability of winning the jackpot Total number of possible outcomes 49 C 6 13 983 816 P Jackpot b 1 0 0000000715 13 983 816 Find the probability of guessing correctly 4 out of 6 numbers Guessing correctly 4 out of 6 numbers two stages 1 …
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