3 - 18.05 Lecture 3 February 7, 2005 n! Pn,k = (n-k)! -...

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18.05 Lecture 3 February 7, 2005 n ! P n,k = ( n - k )! -choose k out of n, order counts, without replacement. n k k out of n, order counts, with replacement. n ! C n,k = k !( n - k )! k out of n, order doesn’t count, without replacement. § 1.9 Multinomial Coefcients These values are used to split objects into groups of various sizes. s 1 , s 2 , ..., s n - n elements such that n 1 in group 1, n 2 in group 2, ..., n k in group k. n 1 + ... + n k = n ± n ²± n - n 1 ²± n - n 1 - n 2 ² × ... ± n - n 1 - ... - n k - 2 ²± n k ² n 1 n 2 n 3 × n k - 1 n k ( n - n 1 - n 2 )! n ! ( n - n 1 )! n 3 !( n - n 1 - n 2 - n 3 )! × ... ( n - n 1 - ... - n k - 2 )! = n 1 !( n - n 1 )! × n 2 !( n - n 1 - n 2 )! × × n k - 1 !( n - n 1 - ... - n k - 1 )! × 1 n ! ± n ² = = n 1 ! n 2 ! ...n k - 1 ! n k ! n 1 , n 2 , ..., n k These combinations are called multinomial coe±cients. Further explanation: You have n “spots” in which you have n! ways to place your elements. However, you can permute the elements within a particular group and the splitting is still the same. must therefore divide out these internal permutations.
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This note was uploaded on 05/02/2011 for the course DYNAM 101 taught by Professor Matuka during the Spring '11 term at MIT.

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3 - 18.05 Lecture 3 February 7, 2005 n! Pn,k = (n-k)! -...

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