Document Sample

Some Sample Choose Numbe rs The Choose Numbers P What is 4 choose 1? P How many w ays are ther e to s elect k objects fr om – 4 n d ist inct objects ? < What is 4 choose 2? – 6 < We called t hat nu mber “ n choose k ” < What is 5 choose 1? < We denoted it n k – 5 P I’ll informa lly call s uc h numbers the “c hoos e < What is 5 choose 2? numbers” – 10 < What is 5 choose 3? – 10 – Not e th at t hi s i s th e s am e as 5 choos e 2. – That’ s becaus e s el ectin g which t hree to pi ck is t he sam e as s el e ctin g whi ch two not to pi ck – And t hus th ere are th e sam e numb ers of ways t o m ake the s el e ctio ns Some Sample Choose Numbers Choose Polynomials P What is 5 choos e 4 < Same as 5 choose 1, which is 5 PFor fixed k, the value of n c hoose k is a PIn gener al, n = n polynomial in n of degree k k n − k PIn gener al, what is n choose 1? PFor example: <n n =1 P What is n choose n? 0 <1 n =n P What is n choose 0? 1 <1 n = (1 / 2) n 2 − ( 1 / 2) n P What is n choose 2? 2 < n(n – 1) / 2, which can be derived from the fact orial n = (1 / 6) n3 − (1 / 2 )n 2 + (1 / 3) n expression for choose numbers 3 Pascal’s Triangle P as cal’s Tri angl e is Binomial Coefficie nts T his t ri angl e cont ain s all o f the de fi ned recursiv el y: choos e num bers. PThe res t of the w orld calls the choos e numbers T here are all 1's on “binomial c oeffic ients ” T o fi nd n choos e k, go to t he th e out si de, and nt h row and s el ect th e kt h ent ry. i nsi de, every PThat is because they appear in the expans ion of ent ry is th e binom ials to integer pow ers s um o f th e B ut bear in min d t hat P as cal’s t ri angl e start s t wo ent ri es P1 above it. wit h t he 0t h row, and Pa + b each row st art s with t he 0t h entry. P a 2 + 2ab + b 2 P... Binomial Coefficients are Choose Numbers Another Question (a + b)5 P How many w ays ar e ther e to place n balls into = two bins, one bin blue and the other bin red, s o (a + b)·(a + b)·(a + b)·(a + b)·(a + b) that exactly k balls go into the blue bin and n – k go into the red bin? In the first expression, t he coefficient of a2 b3, for exa mp le, is w hat w e call a binomial coefficient. n balls T o comp ut e t he coefficient of a2 b3 in the second expression, k balls int o n – k balls int o w e count the number of w ays t o select a t erm from each of the five factors, making sure t hat exactly 3 of them are “ b”. this bin this bin T here are 5 choose 3 w ays t o do t hat. < The answer is n choose k, since w e can simp ly count the number of w ays to s elect t he k for t he blue bin. More Bins Multinomial Coefficients P How many w ays are ther e to plac e n d ist inct objec ts into t d ist inct bins s o that bins < the number of objects in each bin is k 1, k 2, ..., k t objec ts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 < w here k 1 + k 2 + þ+ k t = n? P Can you th ink of a “c hoos e number ” w ay to do P Every rear rangement of the bins g ives another this? ass ignment of objects to bins. P Here is an anagram way to do this: P The number of “bin anagrams” is 16! / 5!5!3!3! bins P In gener al, there ar e n! / k 1!k 2!... k t! objec ts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 P Thes e number are called mult inomial coeff ic ients P Ever y rearrangement of the bins gives another ass ignment of objects to bins. Some Sample Proble ms P How many w ays are ther e to divide tw enty students into four teams, the A, B, C and D teams, each c ontain ing five students? < 20! / 5!5!5!5! = 11,732,745,024 PSix s tudents wer e elected as off ic ers, and the teac her w ishes to ass ign one of those students as pres id ent, tw o as vic e pres idents and three as secretar ies. How many was ar e ther e to do this? < 6!/1!2!3! = 60

DOCUMENT INFO

Shared By:

Categories:

Tags:
Artist's Alley, business cards, NSW Lotteries, fortune telling, room costs, of table, reading Tarot cards, good stick, stick figures, the American

Stats:

views: | 3 |

posted: | 5/28/2011 |

language: | English |

pages: | 3 |

OTHER DOCS BY nyut545e2

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.