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MTH 4150: Midterm 2 Practice 1

Problem 1

What is the coefficient next to \(x^4\) in the power-series expansion of the function \(f(x)=(1+x+x^2+x^3+\cdots)^{10}\)?

Problem 2

Let \(S_n\) be the number of ways to distribute \(n\) identical apples to \(5\) students such that the first two students have to get an even number of apples and the last three students have to get some number of apples divisible by \(3\). Evaluate \(\sum_{n=0}^{\infty}\frac{S_n}{2^n}\).

Problem 3

Find the generating function and the closed form for the sequence \((a_n)_{n=0}^{\infty}\) given by the following recurrence relation: \[a_0=0,\;\;\; a_{n+1}=2a_n+n+5.\]

Problem 4

Find the sequence \((a_n)_{n=0}^{\infty}\) whose exponential generating function is \[H(X)=X+e^{3X}.\]

Problem 5

Consider the sequence \((a_n)_{n=0}^{\infty}\) defined as \(a_n=n\) for \(0\leq n\leq 5\) and \(a_n=0\) for \(n\geq 6\). Let \(F\) be the generating function for the sequence \((a_n)_{n=0}^{\infty}\). Determine \(F(2)\).

Problem 6

Find the number of codewords of length \(5\) from the alphabet \(\{a,b,c,d,e\}\) in which \(b\) occurs an odd number of times.

Problem 7

Let \(n\) be a positive integer. Assume that: \begin{eqnarray*} N_k\mbox{ is the number of pairs }(a,b)\mbox{ of non-negative integers such that } ka+(k+1)b&=&n+1-k. \end{eqnarray*} Find \(N_1+N_2+\cdots+ N_{n+1}\).