CpE602: Applied Discrete Mathematics

1. CpE602: Applied Discrete Mathematics Counting: Binomial Coefficients Generalized Permutations and Combinations Generating Permutations and Combinations

2. 6/11/2012 7:11:16 AM Binomial Coefficients The Binomial Theorem � Gives the coefficients of the expansion of powers of binomial expressions. A binomial is simply the sum of two terms such as x + y. Example: Find the expansion of (x + y)3. Theorem � Let x and y be variables, and let n be a nonnegative integer. Then,

3. 6/11/2012 7:11:16 AM Binomial Coefficients Example: What is the expansion of (x+y)4? Example: What is the coefficient of x12y13 in the expression (x+y)25? [5,200,300] Example: What is the coefficient of x12y13 in the expression (2x-3y)25? Corollary. Let n be a nonnegative integer. Then Corollary. Let n be a positive integer. Then Corollary. Let n be a nonnegative integer. Then

4. 6/11/2012 7:11:16 AM Binomial Coefficients Pascal�s Identity and Triangle Theorem. Let k and n = k be positive integers. Then Pascal�s Identity along with the fact that C(n,0)=C(n,n)=1 can be used to recursively compute binomial coefficients. 1 1 1 Pascal�s 1 2 1 Triangle 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 ��.

5. 6/11/2012 7:11:16 AM Binomial Coefficients Identities of the Binomial Coefficients Theorem (Vandermonde�s Identity). Let m, n and r be nonnegative integers with r not exceeding either m or n. Then Corollary. If n is a nonnegative integer, then Theorem. Let r and n = r be nonnegative integers. Then

6. 6/11/2012 7:11:16 AM Generalized Permutations and Combinations Permutations with repetition Example: How many strings of length r can be formed from the English alphabet? Theorem. The number of r-permutations of a set of n objects with repetition allowed is nr. Combinations with repetition Example: How many ways are there to select four pieces of fruit from a bowl containing apples, oranges and pears if the order in which the pieces are selected does not matter, only the type of the fruit and not the individual piece matters, and there are at least four pieces of each type of fruit in the bowl?

7. 6/11/2012 7:11:16 AM Generalized Permutations and Combinations Example: How many ways are there to select five bills from a cash box containing one, two, five, ten, twenty, fifty and hundred dollar bills? Assume that the order in which the bills are chosen does not matter, that the bills of each denomination are indistinguishable, and that there are at least five bills of each type. Theorem. There are C(n+r-1, r) = C(n+r-1, n-1) r-combinations from a set with n elements when repetition of elements is allowed.

8. 6/11/2012 7:11:16 AM Generalized Permutations and Combinations Example: Suppose that a cookie shop has four different kinds of cookies. How many different ways can six cookies be chosen? Assume that only the type of cookie, and not the individual cookies or the order in which they are chosen, matters. Example: How many solutions does the equation x1+x2+x3 = 11 have, where x1, x2 and x3 are nonnegative integers? Example 6 in the textbook.

9. 6/11/2012 7:11:16 AM Generalized Permutations and Combinations Permutations with indistinguishable objects Example: How many different strings can be made by reordering the letters of the word SUCCESS ? Theorem. The number of different permutations of n objects with n1 indistinguishable objects of type 1, n2 indistinguishable objects of type 2, � , and nk indistinguishable objects of type k, is

10. 6/11/2012 7:11:16 AM Generalized Permutations and Combinations Distributing Objects into Boxes Distinguishable objects and distinguishable objects Example: How many ways are there to distribute hands of 5 cards to each of four players from the standard deck of 52 cards? Theorem. The number of ways to distribute n distinguishable objects into k distinguishable boxes so that ni objects are placed into the ith box equals

11. 6/11/2012 7:11:16 AM Generalized Permutations and Combinations Indistinguishable objects and distinguishable boxes Example: How many ways are there to place 10 indistinguishable balls into eight distinguishable bins? There are C(n+r-1, r) ways to place r indistinguishable objects into n distinguishable boxes. Distinguishable objects and indistinguishable boxes Example: How many ways are there to put four different employees into three indistinguishable offices, when each office can contain any number of employees?

12. 6/11/2012 7:11:16 AM Generalized Permutations and Combinations There is no simple formula for the number of ways to distribute n distinguishable objects into j indistinguishable boxes. A rather complex formula exists: where is a Stirling number of the second kind. Indistinguishable objects and indistinguishable boxes Example: How many ways are there to pack six copies of the same Book into four identical boxes, where a box can contain as many as six books? No simple formula exists.

13. 6/11/2012 7:11:16 AM Generating Permutations and Combinations Generating Permutations Any set of n elements can be place in one-to-one and onto correspondence with {1, 2, 3, � , n}. Hence, we need to construct an algorithm for finding the permutations of {1, 2, � , n}. Lexicographic (or dictionary) ordering. The permutation a1a2�an precedes the permutation b1b3�bn if there is a k such that ai=bi for i < k, and ak < bk. Examples: 23415 precedes 23514 41532 precedes 52143

14. 6/11/2012 7:11:16 AM Generating Permutations and Combinations Given a permutation a1a2�an, we can find the next permutation using the following procedure. - Starting at an, search backwards to find an index j such that aj < aj+1 and aj+1> ak for k = j+2 to n. - Let al be the smallest of {aj+1, � , an} that is greater than aj. The next permutation is a1�aj-1a�ja�j+1�a�n where a�j=al and a�j+1�a�n is constructed by ordering the elements {aj, � , an} - {al} in increasing order. [the pseudocode is given in the textbook] Example: What is the next permutation in lexicographic order after 362541? Example: Generate the permutations of the integers 1, 2 and 3.

15. 6/11/2012 7:11:16 AM Generating Permutations and Combinations Generating combinations Algorithm: Generating the Next Larger Bit String procedure next bit string (bn-1�b1b0: bit string different from 1�11) i := 0 while bi = 1 begin bi := 0 i := i+1 end bi := 1 Example: Find the next bit string after 10 0010 0111.

16. 6/11/2012 7:11:16 AM Generating Permutations and Combinations Algorithm: Generating the Next r-Combination in Lexicographic Order. procedure next r-combination({a1, � , ar}: proper subset of {1, 2, � , n} not equal to {n-r+1, � ,n} with a1<a2<�<ar) i := r while ai = n � r + i i := i - 1 ai := ai + 1 for j := i+1 to r aj := ai + (j - i) Example: Find the next 4-combination of the set {1,2,..,6} after {1,2,5,6}. Example: Generate the 2-combinations of {1,2,3,4}.