3012 MATH Homework Problems Chapter 2, Spring 2009, WTT and MTK
Note: These problems are grouped into two categories. The rst set consists of basic problems that everyone who expects to pass the course should be able to do. The second set consists of challenge problems. Students who would get a C would only be able to do a few of them, while students who would get an A would be able to do allor nearly allof them. Dont be surprised if you are not able to do them allas the course proceeds, we will occasionally slip in a real sleeper that no one on the planet knows how to solve! Group 1: Basic Problems 1. How many 17-letter words can be formed using the 26 lower-case letters of the alphabet if: a. Repetition of letters is allowed. b. Repetition of letters is not allowed. c. Each word contains exactly 4 vowels (the letters a, e, i, o and u), with repetition of letters allowed. d. Each word contains exactly 4 vowels, with repetition of letters not allowed. 2. A newsgroup requires members to login using a password that is a string of length 6, 7 or 8 characters with each character a letter of the English alphabet or a digit from 0 through 9. Also, the website distinguishes between upper and lower case letters. How many dierent passwords are possible? How may passwords are possible if it is required that at least one digit is used? 3. KK Bakery sells 16 varieties of donuts. Coee can be ordered black, with milk, with sugar or with both. a. How may ways can a customer order a coee and a donut? a. How may ways can a customer order a coee and two donuts? b. If a box of donuts contains a dozen donuts, how many dierent boxes could be purchased? 4. How many lattice paths from (0, 0) to (17, 19) pass through (5, 6)? 5. Find the coecient of x7 y 9 in the expansion of (2 + 3xy 5xy 2 )100 and express your answer using binomial coecients. 6. Find the number of integer solutions to the inequality x1 + x2 + x3 + x4 < 126 subject to the following constraints: a. xi > 0 for i = 1, 2, 3, 4. b. xi 0 for i = 1, 2, 3, 4. c. xi 0 for i = 1, 2, 3 and x4 23. d. xi 0 for i = 1, 2, 3 and 0 x4 22.
Then repeat the problem for the inequality x1 +x2 +x3 +x4 126 and the equation x1 +x2 +x3 +x4 = 126. 7. Consider a family with three children (assumed to be distinct) and a collection of 17 identical toys. How many ways to distribute the toys among the three children if a. Each child gets at least one toy. b. The oldest child gets at most 2 toys. c. Suppose the toys are not all the same. In fact, 4 are dolls, 6 are toy trucks and 7 are yoyos. How many ways to distribute the toys? Note: This is actually a problem from the next grouping. It is not a trivial problem. Group 2: Challenge Problems 8. In the last problem from the Basic Section above, suppose the toys are all not the same. In fact, 4 are dolls, 6 are toy trucks and 7 are yoyos. How many ways to distribute the toys? 9. Is 838200020310007224300 a Catalan number? 10. Suppose that a teacher wishes to distribute 25 identical pencils to Ahmed, Barbara, Carlos, and Dieter such that Ahmed and Dieter receive at least one pencil each, Carlos receives no more than ve pencils, and Barbara receives at least four pencils. In how many ways can such a distribution be made? 11. How many integer solutions are there to the equation x1 + x2 + x3 + x4 = 132 provided that x1 > 8, x2 10,x3 0 and x4 < 17? 12. Give a combinatorial argument to prove the identity k n k =n n1 . k1
Hint: Think of choosing a team with a captain. 13. Determine the coecient on x12 y 24 in (x3 + 2xy 2 + y + 3)18 . (Be careful, as x and y now appear in multiple terms!) 14. For each word below, determine the number of rearrangements of the word in which all letters must be used. 1. OVERNUMEROUSNESSES 2. OPHTHALMOOTORHINOLARYNGOLOGY 3. HONORIFICABILITUDINITATIBUS (the longest word in the English language consisting strictly of alternating consonants and vowels 1 )
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http://www.rinkworks.com/words/oddities.shtml
15. How many ways are there to paint a set of 27 elements such that 7 are painted white, 6 are painted old gold, 2 are painted blue, 7 are painted yellow, 5 are painted green, and 0 of are painted red? 16. There are many useful sets that are enumerated by the Catalan numbers. (Volume two of R.P. Stanleys Enumerative Combinatorics contains a famous (or perhaps infamous) exercise in 66 parts asking readers to nd bijections that will show that the number of various combinatorial structures is C(n), and his http://www-math.mit.edu/ rstan/ec/catadd.pdf boasts an additional list of at least 100 parts.) Give bijective arguments to show that each class of objects below is enumerated by C(n). (All three were selected from the list in Stanleys book.) 1. The number of ways to fully-parenthesize a product of n + 1 factors as if the multiplication operation in question were not necessarily associative. For example, there is one way to parenthesize a product of two factors (a1 a2 ), there are two ways to parenthesize a product of three factors ((a1 (a2 a3 )) and ((a1 a2 )a3 )), and there are ve ways to parenthesize a product of four factors: (a1 (a2 (a3 a4 ))), (a1 ((a2 a3 )a4 )), ((a1 a2 )(a3 a4 )), ((a1 (a2 a3 ))a4 ), (((a1 a2 )a3 )a4 ). 2. Sequences of n 1s and n 1s in which the sum of the rst i terms is nonnegative for all i. 3. Sequences 1 a1 an of integers with ai i. For example, for n = 3, the sequences are 111 112 113 122 123. Hint: Think about drawing lattice paths on paper with grid lines and (basically) the number of boxes below a lattice path in a particular column.