Lecture 1.4: Binomial and multinomial coefficients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4190, Discrete Mathematical Structures M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 1 / 8
Motivation � n � The number is called a binomial coefficient, and counts the number of k -element k subsets of an n -element set. The binomial coefficients satisfy a remarkable number of properties. In this lecture, we will explore these, and generalize them to the multinomial coefficients. As a teaser, the entries in Pascal’s triangle are actually binomial coefficients: � 0 � 1 0 � 1 � � 1 � 1 1 0 1 � 2 � � 2 � � 2 � 1 2 1 0 1 2 � 3 � � 3 � � 3 � � 3 � 1 3 3 1 0 1 2 3 � 4 � � 4 � � 4 � � 4 � � 4 � 1 4 6 4 1 0 1 2 3 4 � 5 � � 5 � � 5 � � 5 � � 5 � � 5 � 1 5 10 10 5 1 0 1 2 3 4 5 M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 2 / 8
A recursive identity for binomial coefficients Theorem The binomial coefficients satisfy the following recursive formula: � � � � � � n − 1 n − 1 n = + for all n > 0 and 0 < k < n . , k k − 1 k Proof 1 (algebraic) n ! ( n − 1)! ( n − 1)! Show that k !( n − k )! = ( k − 1)!( n − k )! + � k !( n − k − 1)! . . . Proof 2 (combinatorial) Let’s count, using two different methods, the number of ways to elect k candidates from a pool of n . For the second method, assume that there is one “distinguished” candidate. . . � M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 3 / 8
The binomial theorem We will motivate the following theorem with an example: ( x + y ) 6 = x 6 + 6 x 5 y + 15 x 4 y 2 + 20 x 3 y 3 + 15 x 2 y 4 + 6 xy 5 + y 6 � 6 � x 6 + � 6 � � 6 � x 4 y 2 + � 6 � x 3 y 3 + � 6 � x 2 y 4 + � 6 � xy 5 + � 6 � x 5 y + y 6 . = 0 1 2 3 4 5 6 Theorem For any x , y and n ≥ 1, � � n � n ( x + y ) n = x k y n − k . k k =0 Proof Multiply out, or “FOIL” the product ( x + y )( x + y ) · · · ( x + y ) . � �� � n times This results in 2 n terms, all distinct length- n words in x and y . E.g., for n = 6: xxxxxx + xxxxxy + · · · + xyxyxy + · · · + xxxyyy + · · · + yyyyyy � n � words with exactly k instances of x , so this is the coefficient of x k y n − k . There are k M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 4 / 8
The binomial theorem Corollary � � n n � The n th row of Pascal’s triangle sums to = 2 n . k k =0 Proof 1 (algebraic) Take � � n � n ( x + y ) n = x k y n − k k k =0 and plug in x = y = 1. � Proof 2 (combinatorial) Let’s enumerate the power set of { 1 , . . . , n } of two different ways: (i) Count the number of length- n binary strings (ii) Count the number of size- k subsets, for k = 0 , 1 , . . . , n . � A proof that establishes an identity by counting a carefully chosen set two different ways is called a combinatorial proof. M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 5 / 8
Multinomial coefficients Exercise A police department of 10 officers wants to have 5 patrol the streets, 2 doing paperwork, and 3 at the dohnut shop. How many ways can this be done? � �� �� � 10 5 3 = 10! 5! 3! 10! Answer : 5! 5! · 2! 3! · 3! 0! = 5! 2! 3! = 2520. 5 2 3 This is the same as counting the number of distinct permutations of the word S S S S S P P D D D Definition Suppose that n 1 , . . . , n r are positive integers, and n 1 + · · · + n r = n . Then r − i � � � � � � � � � � � n − n i n n ! n n − n 1 n − n 1 − n 2 := n 1 ! n 2 ! · · · n r ! = · · · i =1 n 1 , n 2 , . . . , n r n 1 n 2 n 3 n r is called a multinomial coefficient. Binomial coefficients are the special case of r = 2. M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 6 / 8
Multinomials and words Consider an alphabet with r letters: { s 1 , . . . , s r } . The number of length- n “words” (i.e., strings) that you can write using exactly n i instances of s i (where n 1 + · · · + n r = n ) is � � n ! n = n 1 ! n 2 ! · · · n r ! . n 1 , n 2 , . . . , n r Examples (i) The number of distinct permutations of the letters in the word MISSISSIPPI is � � 11 11! = 1! 4! 4! 2! = 34650 . 1 , 4 , 4 , 2 (ii) How many length-13 strings can be made using 6 instances of * (“star”) and 7 instances of | (“bar”)? Examples include: *||***||||**| , ******||||||| , |*|*|*|*|*|*| . � � � � 13 = 13! 13 Answer : 6! 7! = = 1716. 6 , 7 6 M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 7 / 8
The multinomial theorem Multinomial coefficients generalize binomial coefficients (the case when r = 2). Not surprisingly, the Binomial Theorem generalizes to a Multinomial Theorem. Theorem For any x 1 , . . . , x r and n > 1, � � n ( x 1 + · · · + x r ) n = � x n 1 1 x n 2 2 · · · x n r r . n 1 , n 2 , . . . , n r ( n 1 ,..., n r ) n 1 + ··· + n r = n M. Macauley (Clemson) Lecture 1.4: Binomial & multinomial coefficients Discrete Mathematical Structures 8 / 8
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