Assignment3-S2019-Solutions.pdf - Introduction to Combinatorics Assignment 3 Solutions University of Toronto Scarborough Due on March 25 on gradescope

Assignment3-S2019-Solutions.pdf - Introduction to...

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Introduction to Combinatorics: Assignment 3,SolutionsUniversity of Toronto ScarboroughDue on March 25 on gradescope by 7pmPlease answer all problems below and your answers on gradescope. Please indicateon gradescope which portion of your work corresponds to which problem.Problem 1.(15 points)Prove that10+21+32+· · ·+n+ 1n=n+ 2n.Solution: Observe that(k+1k)=k+ 1 so the sum is simply the sum of consecutivenatural numbers.Problem 2.(5 points)Relate the digits of the powers of 11: 11,112,113,114with the Pascal triangle.Solution: We write 11 = 10 + 1 and simply use the binomial identity. The coeffi-cients of the power of 10 are (by definition) the digits of the powers of 11.Problem 3.(20 points)Using the theory of linear equations, compute in how many ways can we choosefour non-consecutive numbers from the setT100={1,2,· · ·,100}.For example, thefollowing four numbers{1,2,5,7}cannotbe selected because 1,2 are consecutive.

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