7. Some of the 81920 lattice points in a 40 × 2024 lattice are coloured red. It is known that no four red lattice points are vertices of a rectangle with sides parallel to the axes of the lattice. What is the maximum possible number of red points in the lattice?

Complete a circuit starting at vertex s that passes through vertex u

In an​ orchestra, 23 people can play stringed​ instruments, 21 can play​ brass, and 15 can play percussion.​ Further, 9 of the performers can play both strings and​ brass, whereas 8 can play both strings and percussion. If no one can play all three types of​ instruments, what are the maximum and minimum numbers of people in the​ orchestra? ​(Hint. Consider expressing the numbers in some of the regions of your diagram in terms of a single​ unknown, say​ x.)

16 Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <4 (Exercise 19 in Section 1.4) to prove that every planar graph with less than 12 vertices can be 4-colored.

How many shortest lattice paths start at (5,5) and a. end at (10, 10)? b. end at (10, 10) and pass through (7,6)? c. end at (10, 10) and avoid (7, 6)? Submit answer