Introduction to Combinatorics MATH 239 Spring 2005 Solutions Midterm solutions - Midterm Solutions Math 239 Spring 2005 1[3 Find a closed form

# Introduction to Combinatorics MATH 239 Spring 2005 Solutions Midterm solutions

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Midterm Solutions - Math 239 Spring 2005 1. [3]Find a closed form expression for each of the following formal power series. Youranswer should be expressed in the simplest possible form.Solutions.(a) Geometric series: (b) Binomial Theorem: (c) Geometric series: LetΦ(x) =i0aixi. Define the formal power seriesf(x) =Φ(x)1-x.Determine[xn]f(x)in terms of the coefficientsai. 1
3. [4]LetN={0,1,2,3, . . .}denote the set of nonnegative integers. Define the weightof integernto bew(n) =nifnis odd,n2ifnis even.Determine the generating functionΦN(x)as a rational functionp(x)q(x)wherep(x)andq(x)are polynomials. 4. [8]Letnbe a positive integer and letbndenote the number of compositions ofnintokparts, where each part is one or two. For example,(1,2,1,2,1)and(2,2,1,1,1)are two compositions ofn= 7intok= 5parts.(a)Determine the generating function forbn.