Combinatorial Mathematics : Distinct Elements and Combinations - Four numbers are selected from the set: {-5,-4,-3,-2,-1,1,2,3,4} . In how many ways can the selections be made so that the product ...

5. Four numbers are selected from the set: {-5,-4,-3,-2,-1,1,2,3,4} . In how many ways can the selections be made so that the product of the numbers is positive and: a) The numbers are distinct. b) Each number may be selected as many as four times. c) Each number may be selected at most three times.

Recurrence Relations, Partitions, Generating Functions, Monomials and Noncommutative Algebraic Structure

4. In noncommutative algebra, the term monomial refers to any arrangement of a sequence of variables from a set. For example, in a noncommutative algebraic structure on a set of four variables, {x,y,z,w} , examples of monomials of length 3 are xxx,xyx,xxy,zwy,wzx…….. a) Write a generating function for the number of monomials ...continues

Recurrence Relation : Compound Interest

1. Pauline takes a loan of S dollars at an interest rate of r percent per month, compounded monthly. She plans to repay the loan in T equal monthly installments of P dollars each. a) Let a(subscript n) denote the amount Pauline owes on the loan after n months. Write a recurrence relation for a (subscript n). b) Solve the rec ...continues

Directed Graphs, Vertices and Distinct Paths : Recall that R^3={(x,y,z):x,y,z(subset of R)}. Let G(V,E) be a directed graph, in which V= {(x,y,z)-(subset of R^3) :x,y,z(subset of R),-10<=x,y,z<=10}...

6. Recall that R^3={(x,y,z):x,y,z(subset of R)}. Let G(V,E) be a directed graph, in which V= {(x,y,z)-(subset of R^3) :x,y,z(subset of R),-10<=x,y,z<=10}. Suppose that for any vertex, v=(x,y,z)--[subset of V], the only edges originating at v are the ones joining v to (x+1,y,z),(x,y+1,z),(x,y,z+1) . i.e. any path that originate ...continues

To find the number of monomials of length n, to write a generating function.

Note that the generating function has to be in terms of powers of x. Example: the number of ways to select r balls from a pile of three green, three white, three blue, and three gold balls is the generating function--->(x^0+x^1+x^2+x^3)^4 Here's the problem: 4. In noncommutative algebra, the term monomial refers to any arra ...continues

Prove recurrence by induction

Prove using induction that the recurrence T(n) = T(p*n) + t(q*n) + c*n for n > 1, T(1) = c / (1-p-q), for positive constants c,p,q such that p + q < 1 has the solution: (c*n) / (1-p-q)

Graph Problem with Depth First Search and Breadth First Search

Let G = (V, E) be a connected and undirected graph, and u is a chosen vertex in V. Suppose starting from u, exactly the same tree T is obtained using either breadth first search or depth first search. Prove that G = T, where T is the BFS or DFS tree.

Find the constant term in the expansion (1/(2x^3) + x)^20.

Find the constant term in the expansion of (1/(2x^3) + x)^20. [The constant term is the summand which does not involve any power of x. For example the constant term in 3x^2− 4x + 23 +9x^10 + 5/(3x^6) is 23.] [Note on notation: 2^3 means '2 to the power of 3', so 2^3 = 8] [The pdf file contains the question in pr ...continues

Combinations : How many license plates can a state have if each license plate has 2 letters followed by 4 digits, if the first digit cannot be zero?

How many license plates can a state have if each license plate has 2 letters followed by 4 digits, if the first digit cannot be zero?

Product of Two Infinite Exponential Series

Prove the following product explicitly using infinite series expressions: Here inf = infinity ( Sum_{k=0}^inf u^k/k! ) ( Sum_{l=0}^inf v^l/l! ) = Sum_{m=0}^inf (u+v)^m/m! Please see the attached file for the fully formatted problems.