CMPS 2433 Chapter 8 Counting Techniques Midwestern State University

Review From Previous Chapters 2.6 Binary Search For an ordered list of 2n items, @most n+1 comparisons are needed to find an item For an ordered list of n items, at most log2 n comparisons are needed Example: How many comparisons for list of… 100o items? 1,000,000 items?

Review Permutation: an ordering of a set of elements Permutations of set S with n elements is n! Permutations of r elements taken from n is (n!)/(n-r)! Example - S contains 7 elements How many different permutations? 7! How many permutations of only 5 of the elements? 7!/3!

Review Theorem 1.3: A set of N elements has exactly 2N subsets Consider S = {1, 3, 5, 7, 90} – include or not? Theorem 2.10 (p. 89) :Set S has n elements. # of subsets containing r elements is (n!)/(r! (n-r)!) Referred to as Combinations of n items taken r at a time. C(n,r) Note: r! term eliminates duplicates Example: How many subsets of size 3 from S?

Section 8.2 ~ 3 Fundamental Principles Pigeonhole Principle: If pigeons are placed in pigeon holes and there are more pigeons than holes, then some holes must contain at least 2 pigeons. ~~ If number of pigeons is more than k times the number of holes, then some hole must contain at least k+1 pigeons.

Section 8.2 ~ 3 Fundamental Principles Applications: Pigeonhole Principle How many people must be selected from a collection of 15 couples to ensure at least one couple is selected? How many distinct integers must be chosen to assure there are at least 10 having the same congruence modulo 7? Select any 5 points on the interior of an equilateral triangle having sides length 1. Show that there is at least one pair of points with distance between <= ½.

Fundamental Principle #2 Multiplication Principle: Consider a procedure of k steps. S’pose step 1 can be done in n1 ways, step 2 in n2 ways, etc. The number of different ways the entire procedure can be performed is n1*n2*n3*…*nk.

Fundamental Principle #2 Applications: Multiplication Principle Couple has 5 first names & 3 middle chosen for a baby. How many different baby names? Binary numbers: How many different binary numbers of length 8 are there? What are the values? Phone numbers: How many numbers are possible in the 940 area code? (First 2 digits cannot be 0 or 1) Example 8.10 (p. 410)

Fundamental Principle #3 Addition Principle: Assume k sets with n1 elements in set 1, n2 in set 2, etc. and all elements are distinct. The number of elements in the union of the sets is n1+n2+n3+…+nk Note: Sometimes “solution” is to define the distinct sets so that they can be easily counted.

Fundamental Principle #3 Applications: Addition Principle Couple has 5 girl names and 7 boy names for baby. How many different names? How many integers between 1 – 100 (inclusive) are even or end in 5? Example 8.14 (p. 412)

Homework Section 8.2 Section 8.2 – page 413+ 1 – 36 ~ All except proofs

8.1 Pascal’s Triangle & Binomial Theorem For integers r & n, 1 <= r <= n, C(n,r) = C(n-1,r-1) + C(n-1,r) Example: C (7,5) = C (6,4) + C (6,5) Reminder: C(n,r) = n! / (r! (n-r)!)

Pascal’s Triangle C(0,0) C(1,0) C(1,1) C(2,0) C(2,1) C(2,2) C(3,0) C(3,1) C(3,2) C(3,3) etc…

Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 etc…

Application of Pascal’s Triangle Theorem 8.2: If r and n are integers such that 0 <= r <= n, then C(n,r) = C(n,n-r) Example: C(5,2) = C(5,3)

Theorem 8.3: Binomial Theorem For every positive integer n, (x + y)n = C(n,0)xn + C(n,1)xn-1 y + … + C(n,n-1) x yn-1 + C(n,n)yn C(n,r) are called binomial coefficients

Homework Section 8.1 – page 405+ Problems 1 - 24