Random

In the questions I have below it says a bowl has eight ping pong balls numbered 1,2,2,3,4,5,5,5. You pick a ball at random. a. Find p(the number on the ball drawn is ≥ 3). b. Find p(the number on the ball drawn is even).

Proof by Induction

Show that every positive integer can be written as the product of two numbers. One is the power of 2 and one is odd.

Integers

suppose that integers 1,2,3,4,5,6,7,8,9,10 are arranged randomly along a circle. 1) show that For each circular arrangement, there exists at least three adjacent numbers whose sum is greater than 17 2) take n + 1 integers from {1,2,3,....., 2n}. Show there exist two integers, one divides the other completely.

Divisibility

Suppose A divides N and B divides N. Does this always imply: A * B divides n? Now the question is under what condition A*B will always divide N? Prove it.

Fibonacci Sequence

Let F be the Fibonacci sequence n F = 1, F =1 0 1 F + F n-1 n-2 show 1) For all of n, F = (7/4) ^ n n 2) n+1 n+1 F = 1/√5 ((1+√5) - (1-√5) ) ...continues

Tree Traversal.

Find the: 1. preorder transversal 2. inorder transversal 3. postorder transversal Of the tree attached in the Word document.

More Tree Trans

a. The length of the longest simple circuit in K5 is ???? b. If T is a tree with 999 vertices, then T has ???? edges.

Ordered Pair, Adjacency Matrix and Graph Representation : C4 and W5

With these I need to find an ordered pair, an adjacency matrix, and a graph representation for the graph. a. C4. b. W5.

Edges & Vertices of Kn and Km,n

Please see the attached file for the fully formatted problems. Find Edges & Vertices of Kn and Km,n.

Binary tree

a. Write 3n − (k + 5) in prefix notation: ????. b. If T is a binary tree with 100 vertices, its minimum height is ????. c. Every full binary tree with 50 leaves has ???? vertices.