Illinois State University Mathematics Department

 MAT 312: Probability and Statistics for Middle School Teachers Dr. Roger Day (day@ilstu.edu)

 Counting Problems look at possible solutions

 1 Determine the number of 4-letter arrangements possible with the letters {o,p,s,t}, where no letters are repeated. 2 Solve Problem 1 when repetition is allowed. 3 A teacher plans to write a 5-question test, where the questions are drawn from a test bank of 14 items. Assuming that the order the questions are presented on the test is significant, how many different tests could the teacher create? 4 Four athletes from a team of 12 are to be randomly selected for drug testing. How many different sets of four are possible from the team of 12? 5 A women's soccer conference consists of 8 teams. All teams play each other twice during the season. How many conference games are there in a season? 6 For a multi-state lottery, five numbers are to be picked from 44 different numbers, with no repetition. Assuming that order does not matter, how many different 5-number selections are possible? 7 A computer program selects three numbers from the set {1,2,...,20}, with repetition allowed. (a) How many different 3-number selections can be made? (b) How many of these are sets with no value greater than 5? 8 A field of 12 horses will run the Dentchfield Sweepstakes next week in Millbank. Assuming there are no ties, how many different win-place-show finishes (first-second-third places) are possible? 9 At one of the backstage Oscars parties Sunday night, exactly 55 handshakes took place, where everyone in attendance shook hands with everyone else. How many people were at this party? 10 A ballot on the next county election has 6 referenda on it. If a voter can vote YES, NO, or choose not to vote on each issue, how many different ways can a referenda ballot be marked? 11 Suppose that a fair die is rolled seven times and the result on the face-up side is recorded. How many different ways are there for the seven rolls to result in two 1s, three 2s, and two 6s? Is this a vert likely outcome overall? 12 A seafaring ship carries 10 different signal flags. A specific message is sent by choosing and arranging from two to four different flags. How many different messages can be sent under these conditions? 13 The head table at an annual awards banquet is to have eight speakers seated in a line. (a) How many arrangements are there for those seated at the head table? (b) Suppose the eight were seated at a circular table with eight chairs. Now how many seating arrangements exist? 14 One of many areas where counting strategies are used is in considering the task of reconnecting the wires in a long-distance telephone cable when it has been accidently cut. Suppose a cable contains 120 individual wires that have been cut. (a) How many different ways exist to reconnect the individual wires when the cable is spliced? (b) How much difference would it make in splicing time if the 120 wires in the cable had been grouped into 10 bundles of 12 wires each? 15 Use algebraic reasoning to show that C(n,r) = C(n-1,r) + C(n-1,r-1).