How many shortest lattice paths start at (5,5) and a. end at (13, 13)? b. end at (13, 13) and pass through (8, 7)? c. end at (13, 13) and avoid (8, 7)?

Suppose we used our failing perfect shuffling of digits to mix the digits of the numbers (x,y) that describe a point on the square to get a one-to-one correspondence with the points on a line segment. What point in the square would be paired with the point 0.12000100010001000100010001...?

11. a) Show that the diagonals of a quadrilateral with vertices P(-5, -8), Q(-9, -2), R(-3, 2) and S(1, -4) bisect each other. b) Show that < S is a right angle. 10 8 6- 4- 2- 10-8 -6 -4 -2 -2 4 68 10

Consider a bipartite graph on the set A = {1,2,3,4,5,6} and B = {7,8,9,10,11,12}, where there is an edge between xe A and y e B if and only if x+y is a multiple of 4. What is the size of N({2})? What is the size of N({2,5})? What is the size of N({2,6})? Is there a matching in this graph? %3D