Assignment4-S2019.pdf - Introduction to Combinatorics Assignment 4 University of Toronto Scarborough Due on April 4 on gradescope by 7pm \u2022 Please

# Assignment4-S2019.pdf - Introduction to Combinatorics...

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Introduction to Combinatorics: Assignment 4University of Toronto ScarboroughDue on April 4 on gradescope by 7pmPlease submit your answers on gradescope. For each problem please indicate ongradescope which portion of your work corresponds to this problem.Problem 1.(5 points)An exam with 10 questions is given to 100 students.A student fails the exam ifhe/she gets more than half of the answers wrong. A question is easy if less than half ofthe students get it wrong. Is it possible that all students fail the exam even though allthe questions were easy?Problem 2.(15 points)How many solutions (x1, x2,· · ·, x100) of the inequalityx1+x2+· · ·+x100200are there if all thexiare non-negative integers?Problem 3.(15 points)How many surjective functionsf:ABcan we construct ifA={1,2, ..., n, n+ 1}andB={1,2, ..., n}?Problem 4.(15 points)Prove combinatorially that33+43+53+· · ·+n3=n+ 14.Note: Please present a combinatorial argument. In particular, donotuse induction.

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• Negative and non-negative numbers, The A-Team, Recurrence relation, Generating function, Ahmad, various teams

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