ENGG 2440B Discrete Mathematics for Engineers Tutorial 8

Transcription

1 ENGG 440B Discrete Mathematics for Engineers Tutorial 8 Jiajin Li Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong jjli@se.cuhk.edu.hk November, 018 Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

2 Overview 1 Binomial Coefficient Combinatorial Proof Selected Examples Inclusion-Exclusion (IE) Principle Theorem Selected Examples Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, 018 / 15

3 Binomial Identities Theorem (Pascal s Identity) For any integers n, r 0 with 1 r n 1, ( ) ( ) ( ) n n 1 n 1 +. r r r 1 Theorem (Binomial Theorem) Let n 1 be an integer. For any real numbers x, y, (x + y) n n k0 ( ) n x k y n k. k Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

4 Combinatorial Proof of Pascal s Identity Proof. Counting problem: Let S be the set of all unordered selections of r elements from the n-element ground set {1,,, n}. LHS: S ( n r ). RHS: the sum of two terms, which suggests that the addition principle is at work S S 1 + S. S 1 selections of r elements from {1,,..., n} in which 1 does not appear. S selections of r elements from {1,,..., n} in which 1 appear. Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

5 Combinatorial Proof of Binomial Theorem Proof. Consider the term x k y n k in the expansion of (x + y) n where 0 k n. Such a term can be obtained as follows: we need take an x from k of them ( ) and a y from the remaining n k of them. By definition, there are n ways to perform this task. Hence, the term k ( ) n x k y n k k appears in the expansion of (x + y) n. By summing the above over k 0, 1,..., n, we obtain the desired identity. Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

6 Example Example Let n be a given integer. Give a combinatorial proof of the identity n k ( ) k ( ) n State clearly what you are trying to count and how you are counting it. Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

7 Solution Proof. n k ( ) k ( ) n Counting problem: The RHS is simply choosing 3 numbers out of a set of n + 1 numbers. Out of the 3 numbers, the smallest can be either 1,, 3,, n, n 1. If the smallest number is k, then we can choose numbers from ( k + 1, k +,,) n + 1( to complete ) the set of 3 numbers. n + 1 (k + 1) + 1 n k + 1 There are wayes to do it. Hence we have, ( ) n n 1 ( ) n k + 1 k1 n k ( ) k. Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

8 Example: Vandermonde s Identity Example (Vandermonde s Identity) Give a combinatorial proof of the identity ( ) m + n r r k0 ( ) ( ) m n. k r k Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

9 Solution: Vandermonde s Identity Proof. ( ) m + n r r k0 ( ) ( ) m n k r k Counting problem: The LHS counts the number of ways to choose a committee of r people from a group of m men and n women. The RHS counts the same setting according to cases depending on the number of men on the committees, which can be range from 0 to r. If there are t men, ( ) then there must be r t ( women. ) Since in such a case there are m n wayes to select men and ways to select the women, the t r t ( ) ( ) m n number of such committees is. The result now follows from t r t the rule of sum. Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

10 Inclusion-Exclusion (IE) Principle Theorem In its general form, the principle of inclusion-exclusion states that for finite sets {S 1, S,, S m }, one has the identity S 1 S m S S m S i S j + 1 i<j m 1 i<j<k m S i S j S k + ( 1) m 1 S 1 S S m. Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

11 Example Example Given 3 types of coins, how many ways can one select 0 coins so that no coin is selected more than 8 times? Rough Idea. Step 1: Formulate it as a mathematical counting problem. ( ) k + r 1 x 1 + x + + x r k, x i 0 r 1 Step: Apply the Inclusion-Exclusion (IE) Principle. Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

12 Solution Example Given 3 types of coins, how many ways can one select 0 coins so that no coin is selected more than 8 times? Proof. Step 1: Let x 1, x, x 3 be the number of three different type coins respectively. Then, x 1 + x + x 3 0, 0 x i 8, i 1,, 3. (1) Need: Count the number of integer solution to (1). First, let S be the set of integer solution to Note that S x 1 + x + x 3 0, x i 0, i 1,, 3. ( ) ( ) Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

13 Solution cont d Proof. Step: Remove the solutions in S that violate the upper bounds. Let S i be the set of integer solution to S i x 1 + x + x 3 0, x i 9, x j 0, i j. ( ) S 1 S is the set of integer solution to ( ) 13 78, i 1,, 3. x 1 + x + x 3 0, x 1 9, x 9, x 3 0 Similarly, S 1 S 3 S S 3 6 and S 1 S S 3 0. ( ) Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

14 Solution cont d Proof. By IE principle, Thus, S 1 S S 3 S S 1 S S 3, S 1 S S 3 S 1 + S + S 3 3 S 1 S + S 1 S S 3. S 1 S S 3 ( ) 3 ( ) ( ) Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15

15 Thank you! Jiajin Li (CUHK) ENGG 440B Discrete Mathematics for Engineers Tutorial November 8, / 15