MATC44 2008 - s2 - University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44 Winter 2008 Solutions to Assignment#2

# MATC44 2008 - s2 - University of Toronto at Scarborough...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44, Winter 2008 Solutions to Assignment #2 Problem 23. on page 155: The student needs to cross 24 intersections on her way to school. On each intersection, she has a choice to go to east or to north, but in total she can go only 10 times to east and 14 times to north. (a) There are ( 24 10 ) ways to decide on what intersections she will go to east. On the remaining intersection, she needs to go to north. Hence, she has ( 24 10 ) different ways to school. (b) She needs to cross 9 intersections to get to her friend’s house. Therefore, she has ( 9 4 ) ways to choose on what intersections she will go east, and on remaining she needs to go north. Similarly, there are ( 15 6 ) different ways to get to school from her friend’s house. Therefore, there are ( 9 4 )( 15 6 ) = 15! 4!5!6! different ways from her house to school if she stops at her friend’s house. (c) If they stop in park, they have ( 9 4 )( 9 3 )( 6 3 ) = (9!) 2 (3!) 3 4!5! different ways to school. (d) Since there are ( 9 4 )( 15 6 ) different ways for two of them to get to school together, and ( 9 4 )( 9 3 )( 6 3 ) ways to get to school if they stop in park, there are ( 9 4 )( 15 6 ) - ( 9 4 )( 9 3 )( 6 3 ) ways for two of them to get to school without crossing the intersection where park is. Problem 26. on page 156: 2 n n + 1 + 2 n n = (2 n )! ( n + 1)!( n - 1)! + (2 n )! n ! n ! = = (2 n )! ( n - 1)! n ! 1 n + 1 + 1 n = (2 n )!(2 n + 1) n !( n + 1)! = 2 n + 1 n On the other hand, 1 2 2 n + 2 n + 1 = 1 2 (2 n + 1)! · 2 · ( n + 1) ( n + 1)!( n + 1)! = (2 n + 1)! n !( n + 1)! = 2 n + 1 n . Hence, 2 n n + 1 + 2 n n = 1 2 2 n + 2 n + 1 . Problem 6. on page 200: Let S be the set of all possible packages of 12 chocolate, cinnamon and plain doughnuts. Let A 1 be the set of all 12-doughnuts packs with more than or equal to 7 chocolate doughnuts. Let A 2 be the set of all 12-doughnuts packs with more than or equal to 7 1
cinnamon doughnuts. Finally, let

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