Bookkeeper Rule # perms bo 1 o 2 k 1 k 2 e 1 e 2 pe 3 r = 10! map - PowerPoint PPT Presentation

Mathematics for Computer Science bookkeeper rule MIT 6.042J/18.062J # permutations of the word bookkeeper ? Bookkeeper Rule • # perms bo 1 o 2 k 1 k 2 e 1 e 2 pe 3 r = 10! • map perm o 1 be 1 o 2 k 1 rk 2 e 2 pe 3 to Multinomial Theorem o be o k rk e pe obeokrkepe 10 ! 2 o’s, 2 k’s, 3 e’s: • map is 2!·2!·3!-to-1 2!2!3! Albert R Meyer, April 22, 2013 Albert R Meyer, April 22, 2013 bookkeeper.1 bookkeeper.2 binomial coefficients bookkeeper rule # permutations of length-n binomial a special case : word with n 1 a ’s, n 2 b ’s, …, n k z ’s:    n  n   n! n  k  =   k, n-k    ::=     n 1 ,n 2 , ⋯ ,n k  n 1 !n 2 ! ⋯ n k ! multinomial coefficient Albert R Meyer, April 22, 2013 Albert R Meyer, April 22, 2013 bookkeeper.3 bookkeeper.4 1

multinomials applying the BOOKKEEPER rule What is the coefficient of What is the coefficient of EMS 3 TY EMS 3 TY in the expansion of in the expansion of (E + M + S + T + Y) 7 ? (E + M + S + T + Y) 7 ?   The number of ways to 7   rearrange the letters in  1,1, 3,1,1   the word  SYSTEMS Albert R Meyer, April 22, 2013 Albert R Meyer, April 22, 2013 bookkeeper.6 bookkeeper.7 multinomial coefficients multinomial coefficients What is the coefficient of What is the coefficient of BA 3 N 2 BA 3 N 2 in the expansion of in the expansion of (B + A + N) 6 ? (B + A + N) 6 ?   6 The number of ways to rearrange the letters in   1,3,2 the word   BANANA Albert R Meyer, April 22, 2013 Albert R Meyer, April 22, 2013 bookkeeper.10 bookkeeper.11 2

multinomial coefficients multinomial coefficients Take 14 mile walk including 3 Northward What is the coefficient of miles, 4 Southward, 5 Eastward and 3 r r r 3 r k X 1 X 2 X 3 ⋯ X 1 2 Westward. How many different walks? k in the expansion of = #rearrangements of (X 1 +X +X 3 +…+X k ) n ? N 3 S 4 E 5 W 2 2 ⎛ ⎞   n 14  ⎜  = ⎜   r ,r ,r ,...,r  ⎜ ⎜ 3, 4,5,2    k  ⎝ ⎠ 1 2 3 Albert R Meyer, April 22, 2013 Albert R Meyer, April 22, 2013 bookkeeper.12 bookkeeper.13 The Multinomial Formula multinomial coefficients n   n   X 1 +X 2 +...+ X  =  k   r ,r ,r ,...,r     n ∑ 1 2 3 k r r r 3 ⋯ X r k  X 1 X 2 2 X   2 ,r 3 ,...,r  r + ⋯ +r = n  r 1 ,r 1 3 k k  :: = 0 if r +r 2 +...+r k ≠ n 1 k 1 Albert R Meyer, April 22, 2013 Albert R Meyer, April 22, 2013 bookkeeper.14 bookkeeper.15 3

Preceding slides adapted from: Great Theoretical Ideas In Computer Science Carnegie Mellon Univ., CS 15-251, Spring 2004 Lecture 10 Feb 12, 2004 by Steven Rudich Applied Combinatorics , by Alan Tucker Albert R Meyer, April 22, 2013 bookkeeper.17 4

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