MIT6_042JS10_lec30_sol

MIT6_042JS10_lec30_sol - Massachusetts Institute of...

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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 21, 2010, 763 minutes Solutions to In-Class Problems Week 11, Wed. Problem 1. Find the coefficients of (a) x 5 in (1 + x ) 11 Solution. 11 = 462 5 (b) x 8 y 9 in (3 x + 2 y ) 17 Solution. 17 3 8 2 9 . 8 When (3 x + 2 y ) 17 is expressed as a sum of powers of the summands 3 x and 2 y , the coefficient of (3 x ) 8 (2 y ) 9 is 17 , so the coefficient of x 8 y 9 is this binomial coefficient times 3 8 2 9 . 8 · (c) a 6 b 6 in ( a 2 + b 3 ) 5 Solution. a 6 b 6 = ( a 2 ) 3 ( b 3 ) 2 , so the coefficient is 5 = 10 3 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 .
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