Discrete Mathematics

1) In how many permutations of (1,2,..,8)  are all the even numbers out of place?2) How many permutations of (1,2,..,8) contain none of the patterns 12, 23 and 34?3) The nodes of a 4×82 grid are colored by one of three possible colors, Green, Yellow andBlue. Show that no matter how the nodes are colored, we can always nd two identicallycolored columns.4) Show that, in the sequence 7, 77, 777, 7777, … , there is a number that is divisible by2017.5) A lattice point in R2 is a point (x; y), where both x and y are integers. Prove that ifwe pick 5 lattice points in random, we can always nd 2 points so that their midpoint isalso a lattice point. (The mid point of (x1; y1) and (x2; y2) is given by (x1+y12 ; x2+y22 ).)