Counting. Math 301. November 24, Dr. Nahid Sultana

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1 Basic Principles Dr. Nahid Sultana November 24, 2012

2 Basic Principles Basic Principles The Sum Rule The Product Rule Distinguishable Pascal s Triangle Binomial Theorem

3 Basic Principles Combinatorics: The mathematics of counting.

4 Basic Principles Combinatorics: The mathematics of counting. Question: How many ways can you pick five numbers, from 1 to 10?

5 Basic Principles Combinatorics: The mathematics of counting. Question: How many ways can you pick five numbers, from 1 to 10? Answer: Several ways depending on: 1. Does order matter? 2. Can you repeat numbers?

6 Basic Principles Combinatorics: The mathematics of counting. Question: How many ways can you pick five numbers, from 1 to 10? Answer: Several ways depending on: 1. Does order matter? 2. Can you repeat numbers? Order matter Order not matter Repetition allowed Repetition not allowed

7 Basic Principles The Sum Rule The Product Rule Rule of sum: If a first task can be done in n1 ways, and a second task in n 2 ways, and these tasks cannot be done at the same time, then there are n1 + n 2 ways to do either task

8 Basic Principles The Sum Rule The Product Rule Rule of sum: If a first task can be done in n1 ways, and a second task in n 2 ways, and these tasks cannot be done at the same time, then there are n1 + n 2 ways to do either task Example: Suppose that either a member of a mathematics faculty or a student who has mathematics major is chosen as a representative of a university committee. How many different choices are there for this representative if there are 37 members of the mathematics faculty and 83 mathematics majors?

9 Basic Principles The Sum Rule The Product Rule We can extend the sum rule to more than two tasks.

10 Basic Principles The Sum Rule The Product Rule We can extend the sum rule to more than two tasks. Example: A student can choose a computer project from one of the three list. The three lists contain 23, 15 and 19 possible projects. How many projects are there to choose from?

11 Basic Principles The Sum Rule The Product Rule The Product Rule: Suppose that a procedure can be broken down into two tasks. If there are n 1 ways to do the first task, and n 2 ways to do the second task after the first task has been done then there are n1 n 2 ways to do the procedure.

12 Basic Principles The Sum Rule The Product Rule The Product Rule: Suppose that a procedure can be broken down into two tasks. If there are n 1 ways to do the first task, and n 2 ways to do the second task after the first task has been done then there are n1 n 2 ways to do the procedure. Example: The chair of an auditorium are to be labelled with a letter and a positive integer not exceeding 100. What is the largest number of chairs that can be labelled differently?

13 Basic Principles The Sum Rule The Product Rule The Product Rule: Suppose that a procedure can be broken down into two tasks. If there are n 1 ways to do the first task, and n 2 ways to do the second task after the first task has been done then there are n1 n 2 ways to do the procedure. Example: The chair of an auditorium are to be labelled with a letter and a positive integer not exceeding 100. What is the largest number of chairs that can be labelled differently? We can extend the product rule to more than two tasks.

14 Basic Principles Distinguishable Example: Consider making lists from A, B, C, D, E, F, G. 1. How many length-4 lists are possible if repetition is allowed? 2. How many length-4 lists are possible if repetition is not allowed?

15 Basic Principles Distinguishable Example: Consider making lists from A, B, C, D, E, F, G. 1. How many length-4 lists are possible if repetition is allowed? 2. How many length-4 lists are possible if repetition is not allowed? Theorem: Let A be a set with n elements and 1 r n. Then the number of sequences of length r that can be performed from elements of A, allowing repetition, is n r.

16 Basic Principles Distinguishable Example: Consider making lists from A, B, C, D, E, F, G. 1. How many length-4 lists are possible if repetition is allowed? 2. How many length-4 lists are possible if repetition is not allowed? Theorem: Let A be a set with n elements and 1 r n. Then the number of sequences of length r that can be performed from elements of A, allowing repetition, is n r. Theorem: Let A be a set with n elements and 1 r n. A sequence of r-distinct elements from A can be performed in n(n 1)(n 2)...(n r + 1) ways. This number is written as P(n, r) or npr or P n r, and called the number of permutations of n objects taken r at a time.

17 Basic Principles Distinguishable Theorem: P(n, r) = n! (n r)!.

18 Basic Principles Distinguishable Theorem: P(n, r) = n! (n r)!. When r = n, then P(n, n) = n! (n n)! = n!.

19 Basic Principles Distinguishable Theorem: P(n, r) = n! (n r)!. When r = n, then P(n, n) = n! (n n)! = n!. Therefore, n! is the permutation of n-objects (taken all at a time).

20 Basic Principles Distinguishable Theorem: P(n, r) = n! (n r)!. When r = n, then P(n, n) = n! (n n)! = n!. Therefore, n! is the permutation of n-objects (taken all at a time). Example: A = {a, b, c}. Example: How many words of three distinct letters can be formed from the letters of the word MAST?

21 Basic Principles Distinguishable Theorem: The number of distinguishable permutations that can be formed from a collection of n-objects where the first objects appears k1 times, the second object k2 times, and so on, is n! P(n, k 1, k 2,..., k r ) = k 1!k 2!...k r!.

22 Basic Principles Distinguishable Theorem: The number of distinguishable permutations that can be formed from a collection of n-objects where the first objects appears k1 times, the second object k2 times, and so on, is n! P(n, k 1, k 2,..., k r ) = k 1!k 2!...k r!. Example: How many distinguishable permutations of the letters in the word BANANA are there?

23 Basic Principles Question: How many subsets can be made by selecting r elements from a set with n elements? Repetition not allowed and order doesn t matter.

24 Basic Principles Question: How many subsets can be made by selecting r elements from a set with n elements? Repetition not allowed and order doesn t matter. Theorem: Let A be a set with A = n, and 1 r n. Then the number of combinations of the elements of A, taken r at a time where order does not count, is C(n, r) = ( n r ) = This is simply a r-element subset of A. n! r!(n r)!.

25 Basic Principles Question: How many subsets can be made by selecting r elements from a set with n elements? Repetition not allowed and order doesn t matter. Theorem: Let A be a set with A = n, and 1 r n. Then the number of combinations of the elements of A, taken r at a time where order does not count, is C(n, r) = ( n r ) = This is simply a r-element subset of A. Therefore, C(n, r) = P(n,r) r!. n! r!(n r)!.

26 Basic Principles Example: How many 4-element subsets does A = {1, 2, 3, 4, 5, 6, 7, 8, 9} have?

27 Basic Principles Example: How many 4-element subsets does A = {1, 2, 3, 4, 5, 6, 7, 8, 9} have? How many ways are there to select 5 players from a 10-members tennis team to make a trip to a match at another school?

28 Basic Principles Pascal s Triangle Binomial Theorem The numbers ( n r ) are called a binomial coefficients. The name binomial coefficient is used because these numbers occur as coefficients in the expansion of powers of binomial expression such as (a + b) n.

29 Basic Principles Pascal s Triangle Binomial Theorem The numbers ( n r ) are called a binomial coefficients. The name binomial coefficient is used because these numbers occur as coefficients in the expansion of powers of binomial expression such as (a + b) n. Let n and r be non-negative integers and r n then ( ) ( ) n n = n r r

30 Basic Principles Pascal s Triangle Binomial Theorem The numbers ( n r ) are called a binomial coefficients. The name binomial coefficient is used because these numbers occur as coefficients in the expansion of powers of binomial expression such as (a + b) n. Let n and r be non-negative integers and r n then ( ) ( ) n n = n r r ( n i ) = 0 for i < 0 or i > n, where n is a positive integer.

31 Basic Principles Pascal s Triangle Binomial Theorem Pascal s Identity: Let n and r be non-negative integers and r n then ( ) ( ) ( ) n + 1 n n = + r r 1 r

32 Basic Principles Pascal s Triangle Binomial Theorem Pascal s Identity: Let n and r be non-negative integers and r n then ( ) ( ) ( ) n + 1 n n = + r r 1 r Pascal s triangle: Triangular array of the binomial coefficients. ( ) 0 1. Top row (row 0): 0 ( ) 1 2. Next row (row 1): listing the value of with k = 0, 1 k ( ) 2 3. Next row (row 2): listing the value of with k = 0, 1, 2 k 4. and so on...

33 Basic Principles Pascal s Triangle Binomial Theorem In general, each row listing the numbers ( ) n + 1 a row listing the numbers. k ( n k ) is just above

34 Basic Principles Pascal s Triangle Binomial Theorem ( n k ) is just above In general, each row listing the numbers ( ) n + 1 a row listing the numbers. k ( ) n + 1 Any number in this triangle falls immediately k below and between the positions of two numbers ( ) n and in the previous row. k ( n k 1 Therefore, by Pascal s identity, any number (other than 1) is the sum of the two numbers immediately above it. )

35 Basic Principles Pascal s Triangle Binomial Theorem ( n k ) is just above In general, each row listing the numbers ( ) n + 1 a row listing the numbers. k ( ) n + 1 Any number in this triangle falls immediately k below and between the positions of two numbers ( ) n and in the previous row. k ( n k 1 Therefore, by Pascal s identity, any number (other than 1) is the sum of the two numbers immediately above it. More in class... )

36 Basic Principles Pascal s Triangle Binomial Theorem The n-th row of the Pascal s triangle lists the coefficients of (x + y) n. For example, for n = 2, we have (x + y) 2 = 1x 2 + 2xy + 1y 2, and Row 2 of Pascal s triangle lists the coefficients

37 Basic Principles Pascal s Triangle Binomial Theorem The n-th row of the Pascal s triangle lists the coefficients of (x + y) n. For example, for n = 2, we have (x + y) 2 = 1x 2 + 2xy + 1y 2, and Row 2 of Pascal s triangle lists the coefficients This is true for every n, and the result is known as the binomial theorem.

38 Basic Principles Pascal s Triangle Binomial Theorem The n-th row of the Pascal s triangle lists the coefficients of (x + y) n. For example, for n = 2, we have (x + y) 2 = 1x 2 + 2xy + 1y 2, and Row 2 of Pascal s triangle lists the coefficients This is true for every n, and the result is known as the binomial theorem. Binomial Theorem: If n is a non-negative integer, then (x + y) n = n k=0 ( n k ) x n k y k.

39 Basic Principles Pascal s Triangle Binomial Theorem The n-th row of the Pascal s triangle lists the coefficients of (x + y) n. For example, for n = 2, we have (x + y) 2 = 1x 2 + 2xy + 1y 2, and Row 2 of Pascal s triangle lists the coefficients This is true for every n, and the result is known as the binomial theorem. Binomial Theorem: If n is a non-negative integer, then Examples: In class. (x + y) n = n k=0 ( n k ) x n k y k.

40 Basic Principles : If n pigeonholes are occupied by n + 1 or more pigeons, then at least one pigeonhole is occupied by more than one pigeon.

41 Basic Principles : If n pigeonholes are occupied by n + 1 or more pigeons, then at least one pigeonhole is occupied by more than one pigeon. Suppose there are 8 people in a group, then at least two of them were born on the same day of the week.

42 Basic Principles : If n pigeonholes are occupied by n + 1 or more pigeons, then at least one pigeonhole is occupied by more than one pigeon. Suppose there are 8 people in a group, then at least two of them were born on the same day of the week. Generalized : If n pigeonholes are occupied by kn + 1 or more pigeons, where k is a positive integer, then at least one pigeonhole is occupied by k + 1 or more pigeons.

43 Basic Principles : If n pigeonholes are occupied by n + 1 or more pigeons, then at least one pigeonhole is occupied by more than one pigeon. Suppose there are 8 people in a group, then at least two of them were born on the same day of the week. Generalized : If n pigeonholes are occupied by kn + 1 or more pigeons, where k is a positive integer, then at least one pigeonhole is occupied by k + 1 or more pigeons. Find the minimum number of students in a class to be sure that three of them are born in the same month.

Counting Methods. CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo

Counting Methods. CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo Counting Methods CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 48 Need for Counting The problem of counting