w13 - 1. Find the coecient of x 12 in 1 (1+ x ) 8 2. Find a...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Warmup - Do these together! 1. How many ways are there to rearrange the letters of the English alphabet so that “CAT,” “DOG,” and “FISH” are all absent. 2. What sequence is represented by each of the following generating functions? (a) ( x 2 + 1) 3 (b) 1 (1 - 2 x 2 ) (c) x 9 - 1 x - 1 Inclusion-Exclusion 1. Find the number of solutions in non-negative integers to x 1 + x 2 + x 3 = 15 where x 1 6 , x 2 8 , x 3 3 2. How many positive integers not exceeding 1000 are coprime to 84? Generating Functions
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Unformatted text preview: 1. Find the coecient of x 12 in 1 (1+ x ) 8 2. Find a generating function for the number of solutions in non-negative integers to x 1 + x 2 + x 3 = k where 3 x 1 , 2 x 2 10 , 5 x 3 . What if you add the requirement that x 3 is even? 3. Find a generating function for the number of ways to make n cents using pennies, nickels, dimes, and quarters, where the order of the coins doesnt matter. Use this to nd the number of ways to make 10 cents....
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