91Ó°ÊÓ

For each of the following, determine a generating function and indicate the coefficient in the function that is needed to solve the problem. (Give both the polynomial and power series forms of the generating function, wherever appropriate.) Find the number of integer solutions for the following equations: a) \(c_{1}+c_{2}+c_{3}+c_{4}=20,0 \leq c_{i} \leq 7\) for all \(1 \leq i \leq 4\) b) \(c_{1}+c_{2}+c_{3}+c_{4}=20,0 \leq c_{1}\) for all \(1 \leq i \leq 4\), with \(c_{2}\) and \(c_{3}\) even c) \(c_{1}+c_{2}+c_{3}+c_{4}+c_{5}=30,2 \leq c_{1} \leq 4\) and \(3 \leq c_{r} \leq 8\) for each \(2 \leq i \leq 5\) d) \(c_{1}+c_{2}+c_{3}+c_{4}+c_{3}=30,0 \leq c_{i}\) for all \(1 \leq i \leq 5\), with \(c_{2}\) even and \(c_{3}\) odd

Find all partitions of \(7 .\)

a) Find the generating function for the number of ways to select 10 candy bars from large supplies of six different kinds. b) Find the generating function for the number of ways to select, with repetitions allowed, \(r\) objects from a collection of \(n\) distinct objects.

Find the generating function for the number of integer solutions to the equation \(c_{1}+\) \(c_{2}+c_{3}+c_{4}=20\) where \(-3 \leq c_{1},-3 \leq c_{2},-5 \leq c_{3} \leq 5\), and \(0 \leq c_{4}\).

a) Find the coefficient of \(x^{7}\) in \(\left(1+x+x^{2}+x^{3}+\cdots\right)^{15}\). b) Find the coefficient of \(x^{7}\) in \(\left(1+x+x^{2}+x^{3}+\cdots\right)^{n}\) for \(n \in \mathbf{Z}^{+}\).

How many 10-digit telephone numbers use only the digits \(1,3,5\), and 7 , with each digit appearing at least twice or not at all?

For integers \(n, k \geq 0\) let \- \(P_{1}\) be the number of partitions of \(n\). \- \(P_{2}\) be the number of partitions of \(2 n+k\), where \(n+k\) is the greatest summand. \- \(P_{3}\) be the number of partitions of \(2 n+k\) into precisely \(n+k\) summands. Using the concept of the Ferrer's graph, prove that \(P_{1}=P_{2}\) and \(P_{2}=P_{3}\), thus concluding that the number of partitions of \(2 n+k\) into precisely \(n+k\) summands is the same for all \(k\).

Using a Ferrer's graph, show that the number of partitions of an integer \(n\) into summands not exceeding \(m\) is equal to the number of partitions of \(n\) into at most \(m\) summands.

If a 20 -digit ternary \((0,1,2)\) sequence is randomly generated, what is the probability that: (a) It has an even number of 1 's? (b) It has an even number of 1 's and an even number of 2 's? (c) It has an odd number of 0 's? (d) The total number of 0 's and l's is odd? (c) The total number of 0 's and l's is even?

Two cases of soft drinks, 24 bottles of one type and 24 of another, are distributed among five surveyors who are conducting taste tests. In how many ways can the 48 bottles be distributed so that each surveyor gets (a) at least two bottles of each type? (b) at least two bottles of one particular type and at least three of the other?