HW 4 - Principle of Inclusion-Exclusion (Due Mon 10/8) 1. A six-sided die is rolled repeatedly until the numbers one through five have appeared at least once each. What is the probability that this...

Please do all the problems in the attached file


HW 4 - Principle of Inclusion-Exclusion (Due Mon 10/8) 1. A six-sided die is rolled repeatedly until the numbers one through five have appeared at least once each. What is the probability that this happens in the first n rolls ? Use a calculator to find the first n where the probability exceeds 0.5 2. Suppose that n soldiers march in a single column such that everyone (except the front one) can see only the person in front of him/her. How many ways are there to arrange them (in a single column) the next day such that everyone sees someone different than the one they saw today ? 3. A mathematics department has n professors and 2n courses, two assigned to each professor each semester. How many ways are there to assign them in the Spring so that no professor teaches the same two courses in the Spring as in the Fall ? 4. Use PIE to prove that ∑ (−1)k ( n k ) (n− k)n = n! 5. Evaluate the sum below using PIE (i) n∑ k=0 (−1)k ( n k ) (n− k) (ii) m∑ k=0 (−1)k ( n k )( n− k m− k ) 6. Evaluate n∑ i=0 (−1)i ( n i ) ik, when 0 ≤ k < n="" or="" k="n" or="" k=""> n. 1