CMSC 250 Jerry Alan Fails Due Tuesday July 17 2007 HW 11 Student ID You must work alone on your homework and homework must be written legibly single sided on your own lined paper or typed with the answers clearly labeled and in the sequential order as assigned You must write your name and university ID number in the upper right hand corner of your homework Staple all pages together and be sure that your name appears on every sheet 1 10 points if wrong Write your name clearly on each page Write the time and place of the Final Exam 2 20 points How many permutations of abcde are there in which the first character is a b or c and the last character is c d or e 3 10 points If n is a positive integer how many 5 tuples of integers from 1 through n can be formed in which the elements of the 5 tuple are written in decreasing order but are not necessarily distinct In other words how many 5 tuples of integers h i j k m are there with n h i j k m 1 4 10 points Assuming that all years have 365 days and all birthdays occur with equal probability how large must n be so that in any randomly chosen group of n people the probability that two or more have the same birthday is at least This is called the birthday problem Many find the answer surprising 5 10 points For how many integers from 1 through 99 999 is the sum of their digits equal to 9 6 10 points For each of the following write the specified term of the expansion Simplify all coefficients so the denominators do NOT have a factorial a Write and simplify the 5th term of the expansion of 3 x 2 y 10 b Write and simplify the 10th term of the expansion of x y 13 7 8 points Let A 1 2 3 4 5 P A be the power set of A Define a function F P A Z as follows For all sets X in P A 0 if X has an even number of elements F X 1 if X has an odd number of elements Find the following a F 1 3 4 b F 2 3 c F d F 2 3 4 5 HW 11 Due Tuesday July 17 2007 Page 1 of 2 CMSC 250 Jerry Alan Fails Due Tuesday July 17 2007 HW 11 Student ID 8 8 points Determine the following a How many one to one functions are there from a set with three elements to a set with three elements b How many one to one functions are there from a set with m elements to a set with n elements where m n c How many onto functions are there from a set with three elements to a set with five elements d How many onto functions are there from a set with four elements to a set with three elements 9 8 points Function f is defined below on a set of real numbers Determine whether or not f is one to one and justify your answer f x x 1 for all real numbers x 1 x 1 10 8 points Let P A be the power set of A Define F P a b c Z as follows For all A in P a b c F A the number of elements in A a Is F one to one Prove or give a counterexample b Is F onto Prove or give a counterexample 11 8 points Let S be the set of all strings in a s and b s and define C S S by C s as for all s S C is called concatenation by a on the left a Is F one to one Prove or give a counterexample b Is F onto Prove or give a counterexample 12 No points will be awarded for this assignment unless this is done Sign your name to the following honor code statement I pledge on my honor that I have not given or received any unauthorized assistance on this assignment HW 11 Due Tuesday July 17 2007 Page 2 of 2
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