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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

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posted: | 5/28/2011 |

language: | English |

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