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

Assignment4-S2019-solutions.pdf - Introduction to...

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Introduction to Combinatorics: Assignment 4 University of Toronto Scarborough Due on April 4 on gradescope by 7pm Please submit your answers on gradescope. For each problem please indicate on gradescope 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 if he/she gets more than half of the answers wrong. A question is easy if less than half of the students get it wrong. Is it possible that all students fail the exam even though all the questions were easy? Solution: Let’s doubly count the number of wrong answers. If all 100 students failed then each of them got at least 6 problems wrong and hence there should have been at least 600 wrong answers. If all questions were easy then at most 50 students gave a wrong answer for each of them and hence there should have been at most 500 wrong answers. Contradiction. Problem 2. (15 points) How many solutions ( x 1 , x 2 , · · · , x 100 ) of the inequality x 1 + x 2 + · · · + x 100 200 are there if all the x i are non-negative integers? Solution: We introduce a new variable x 101 such that x 1 + x 2 + · · · + x 100 + x 101 = 200 Now we can associate any solution ( x 1 , x 2 , .... , x 100 ) of the original inequality to the solution ( x 1 , x 2 , ..., x 100 , x 101 ) of the above equality. The above equality admits 101 - 1 + 200 200 different solutions. Problem 3. (15 points) 1
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How many surjective functions f : A B can we construct if A = { 1 , 2 , ..., n, n + 1 } and B = { 1 , 2 , ..., n } ?
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  • Fall '08
  • Elementary arithmetic, Grammatical number, Recurrence relation, Even and odd functions, Generating function

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