exam2 - AMS 301 Finite Mathematical Structures Ning SUN...

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Unformatted text preview: AMS 301 Finite Mathematical Structures Ning SUN Exam 2: Aug 11 2009, 6:00-7:00 pm Write all your answers on the answer sheet, using the backs of pages if needed. This exam is CLOSED BOOK and CLOSED NOTES. Please show all your work to get the full credit. Choose FIVE of the seven questions, the total is 110. 1. (20 pt) Consider the problem of counting the ways to select 25 objects from 8 types with at least 2 objects of each type. (a) (6 pt) Model this problem as an integer-solution-of-an-equation problem. (b) (7 pt) Model this problem as a certain coecient of a generating function. (c) (7 pt) Solve this problem. 2. (20 pt) What is the probability that a 4-card subset out of the 52-card deck has no pairs (each card has a dierent value)? 3. (20 pt) How many arrangements of MATHEMATICAL that do NOT contain AE? 4. (20 pt) Build a generating function for the following problems (You do NOT need to calculate the coecient): (a) (10 pt) The number of election outcomes in the race for class president if there are 4 candidates and 30 votes. (b) (10 pt) The number of ways to give a total of 87 cents, using one dollar worth of pennies (1 cent), one dollar worth of nickels (5 cents) and one dollar worth of dimes (10 cents) 5. (20 pt) How many ways are there to form a sequence of 10 letters from 4A's, 4B's, 4C's, 3D's if each letter must appear at least twice? 6. (30 pt) Find the corresponding coecients: (a) (10 pt) What is the coecient of (b) (20 pt) What is the coecient of x20 in the expansion of (x + x2 + x3 + x4 + · · ·)3 x10 in the expansion of (1 − x4 )5 (1 + x + x2 + x3 + · · ·)6 ? 7. (30 pt) How many arrangements of the letters in INCONSISTENT have ALL of the following properties: (a) The vowels are non-consecutive; (b) The consonants are NOT in alphabetical order; (c) The arrangement ends with a vowel. 1 ...
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