Homework assignment Number 3Math 145 // due on 01/26/20071. Problems Section 3.8: 6, 8, 9, 10, 13, 15.2. A “walk” starts at the origin and makes up (resp. down) steps (1,1) (resp. (1,-1)).(a) How many walks are there from the origin to (p, q) for given integersp >0,q?(b) Suppose now that the walk starts atA= (0,0) and finishes atB= (2n,0). We wantto determine the number of such walks that go below thex-axis before 2n. The aim isto show that this number is equal to the number of walks starting atAand ending atC= (2n,-2).i. LetWbe a walk going below thex-axis.Suppose thatWreaches the level-1for the first time at the instantn <2n. SetW1to be the trajectory which is thereflection with respect to the axisy=-1 ofW. Make a plot of two such trajectoriesWandW1. What are the endpoints ofW1?ii. Consider then the walkWwhich first follows the trajectory ofWuntil timenandthen follows the same trajectory asW1until time 2n. Show thatW→Wdefinesa bijection between the set of walks fromAtoBgoing below the
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