Multiplication Principle, Permutations, and Combinations
... In Example 1, we see that there are two possible outcomes from the first operation of flipping a coin and six possible outcomes from the second operation of throwing a die. Hence, by the multiplication principle, there are 2 6 12 possible combined outcomes of flipping a coin followed by throwing ...
... In Example 1, we see that there are two possible outcomes from the first operation of flipping a coin and six possible outcomes from the second operation of throwing a die. Hence, by the multiplication principle, there are 2 6 12 possible combined outcomes of flipping a coin followed by throwing ...
Chapter 1 Lecture Notes
... Step 1 Understand the problem. We are looking for a number that goes through a series of changes to turn into 60. Step 2 Devise a plan. Work backwards to undo the changes. Step 3 Carry out the plan. The final amount was 60. Divide by 4 to undo quadruple = 15. Add 47 to get 62, then divide by 2 = 31. ...
... Step 1 Understand the problem. We are looking for a number that goes through a series of changes to turn into 60. Step 2 Devise a plan. Work backwards to undo the changes. Step 3 Carry out the plan. The final amount was 60. Divide by 4 to undo quadruple = 15. Add 47 to get 62, then divide by 2 = 31. ...
Comp 205: Comparative Programming Languages
... Lazy Evaluation and Infinite Lists Lecture notes, exercises, etc., can be found at: www.csc.liv.ac.uk/~grant/Teaching/COMP205/ ...
... Lazy Evaluation and Infinite Lists Lecture notes, exercises, etc., can be found at: www.csc.liv.ac.uk/~grant/Teaching/COMP205/ ...
CS311H: Discrete Mathematics Cardinality of Infinite Sets and
... Now, we’ll create a new real number R and show that it is not equal to any of the Ri ’s in this sequence: ...
... Now, we’ll create a new real number R and show that it is not equal to any of the Ri ’s in this sequence: ...
3. Complex Numbers
... ba. Octonions are eight-component numbers which are both non-commutative and nonassociative: (ab)c 6= a(bc). These and other still-more-complicated number systems have some applications in physics and other fields, but are overall much less important than C. Another reason complex numbers are so mat ...
... ba. Octonions are eight-component numbers which are both non-commutative and nonassociative: (ab)c 6= a(bc). These and other still-more-complicated number systems have some applications in physics and other fields, but are overall much less important than C. Another reason complex numbers are so mat ...
Invariants and Algorithms 1 Introduction 2 Examples
... 3. (Putnam 2010) There are 2010 boxes labeled B1 , . . . , B2010 , and 2010n balls have been distributed among them, for some positive integer n. You may redistribute the balls by a sequence of moves, each of which consists of choosing an i and moving exactly i balls from Bi into any other box. For ...
... 3. (Putnam 2010) There are 2010 boxes labeled B1 , . . . , B2010 , and 2010n balls have been distributed among them, for some positive integer n. You may redistribute the balls by a sequence of moves, each of which consists of choosing an i and moving exactly i balls from Bi into any other box. For ...
