MIT6_042JS10_lec30_prob

MIT6_042JS10_lec30_prob - that your CFO’s formula agrees...

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�� Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science April 21 Prof. Albert R. Meyer revised April 16, 2010, 1284 minutes In-Class Problems Week 11, Wed. Problem 1. Find the coefficients of (a) x 5 in (1 + x ) 11 (b) x 8 y 9 in (3 x + 2 y ) 17 (c) a 6 b 6 in ( a 2 + b 3 ) 5 Problem 2. You want to choose a team of m people for your startup company from a pool of n applicants, and from these m people you want to choose k to be the team managers. You took 6.042, so you know you can do this in �� n m m k ways. But your CFO, who went to Harvard Business School, comes up with the formula n n k . k m k Before doing the reasonable thing —dump on your CFO or Harvard Business School —you decide to check his answer against yours. (a) Give a combinatorial
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Unformatted text preview: that your CFO’s formula agrees with yours. (b) Verify this combinatorial proof by giving an algebraic proof of this same fact. Problem 3. (a) Now give a combinatorial proof of the following, more interesting theorem: n ± ² n n 2 n − 1 = k (1) k k =1 Hint: Let S be the set of all length-n sequences of 0’s, 1’s and a single *. (b) Now prove ( 1 ) algebraically by applying the Binomial Theorem to (1+ x ) n and taking deriva-tives. Creative Commons 2010, Prof. Albert R. Meyer . MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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