Please write in process and answers in a separate piece of paper. Additional $52 as tip once the document is uploaded in 8 hours.

1.7. 1.8. [9. 20. 21. 22. 23. 24. 26. 27. 28. 29. 30. 31. 32. 33. (a) Find number oi strings in symbols 0.1.2.3.4 of length 14 that contain 12 consecutive 0's: (b) Same question for bit strings. Find the number of Ni integers r such that (a) 1 5 z 5 1.00. .1 is divisible by 2 or a or 5: (b) 71.00 5: 5100. a: is asquale (ofan integer) or: isdivisible by5 or 7. Apply the inclusion-exclusion formula to find the number ofall surjective Functions f ' B > C such that (n) IBI = m. ICI = n: (b) B = {1.2. a. 4. 5.6.7.8}. C = {1.2.3.4. 5.6} and f(1)=1: (c) B = {1.2.3.4.5.a.7.e}. C= {1.2.33.5} and ?) =4J{E) = 4- (a) Compute the number of all d'mdbutions of 7 distinct toys among 5 children so that every child gets at least one toy. (b) Compute the number ufall letter strings of length E in letters A. B, C. D that contain all four letters. Suppose that :1 people check their hats and later get them back. at random. Compute the probability that no one gets hisfher own hat. Find out which of reexive. symmetric. transitive properties hold for given below relation R on the set A. is R an equivalence relation? I\" so. nd equivalence clam at R and compute their cardinalitia. (a)A=Zi R=ilwayl l |''-y|0};(C) A=z- R=~nyl|=59ii (d) A=Z.R={{z.y}l |$|=|y|}: (d) A=Z.R={{:.y] |=+39555VEDP Find out which of reexive. symmetric, transitive properties hold [or the given below relation R on the set A. Is R an equivalence relation? If so. nd equivalence classes of R and compute their cardinalities (a) A =2. R= shy) l:z:73y is even}: (b) A = {z | z is abit string of length n}. R = {{zz:.y)| thenumbers od'O's in z and yare equal}: (c) A = {c | c is abit string of length 3}. R = {(3.30 | the dierence between the numbers of 1's in :z: and y is even}. Find the number ofall equivalence relations R on a set A if (a) IAI = 3: (b) lAI = 4: (c) |A| = 5 and R has two equivalence classes: [6} |A| = 5. . Compute the number of all equivalence relations on the set A. = {1,2, 3. 4. 5} such that (a) theyr contain precisely three equivalence classa; (b) 1 and 2 are not related; (e) 1. 2. and 3 are not pairwise related. Give examples of such equivalence relations. (1:) Construct the adjacency matrices for graphs K... K33 and W... (b) Construct. if possible. a simple graph whose adjacency matrix has a column consisting entirely of 0's. (c) Construct. if possible, a simple graph whom adjacency matrix has a column consisting entirely of 1's. Count the numbers of verticis and edges in graphs W... C... K... Km... Q... Determine which of the following graphs are bipartite: W... C... K... Km... Q... Draw all nonisclnrorphic simple graphs with (a) 3 vertica; (b) 4 vertices; (c) 5 vertices and 3 edges. Draw all nenisoanorphic simple connected graphs with (a) 4 edges: {b} 5 edges: (c) 6 edges and 4 vertic. Find the number of Ni paths of length 2. 3. 4 between two adjacent vertices v1.1}; in (a) K]; (b) Ca: (6) Km: [5} We- Are there Euler. Hamilton circuits in (a) Q"; (b) K... (c) Kym? Are there Euler. Hamilton paths in (a) C": (b) Kn; (C) Kinn