Discrete Mathematics and Its Applications
Discrete Mathematics and Its Applications
8th Edition
ISBN: 9781260501759
Author: ROSEN
Publisher: MCG
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Chapter 6.4, Problem 26E

Prove the identity (")(') = (J)(Xf), whenever n, r, andkare nonnegative integers withr< n and k

  1. using a combinatorial argument.

  • using an argument based on the formula for the number of r-combinations of a set withnelements

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    Chapter 6 Solutions

    Discrete Mathematics and Its Applications

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Show that in any set...Ch. 6.2 - Prob. 10ECh. 6.2 - Prob. 11ECh. 6.2 - Prob. 12ECh. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Show that if five integers are selected from the...Ch. 6.2 - i6. Show that if seven integers are selected from...Ch. 6.2 - How many numbers must be selected from the set...Ch. 6.2 - Howmany numbers must be selected from the set...Ch. 6.2 - A company stores products in a warehouse. Storage...Ch. 6.2 - Suppose that there are nine students in a discrete...Ch. 6.2 - i. Suppose that every student in a discrete...Ch. 6.2 - Prob. 22ECh. 6.2 - Construct a sequenceof16 positive integers that...Ch. 6.2 - Prob. 24ECh. 6.2 - Show that whenever 25 girl? and 25 boys are seated...Ch. 6.2 - Prob. 26ECh. 6.2 - Descnbe an algorithm in pseudocode for producing...Ch. 6.2 - Prob. 28ECh. 6.2 - Prob. 29ECh. 6.2 - Prob. 30ECh. 6.2 - Prob. 31ECh. 6.2 - Prob. 32ECh. 6.2 - Prob. 33ECh. 6.2 - Prob. 34ECh. 6.2 - In the 17th century, there were more than 800,000...Ch. 6.2 - Prob. 36ECh. 6.2 - Prob. 37ECh. 6.2 - Prob. 38ECh. 6.2 - A computer network consists of six computers, Each...Ch. 6.2 - Prob. 40ECh. 6.2 - Prob. 41ECh. 6.2 - Prob. 42ECh. 6.2 - Ad arm wrestler is the champion for a period of 75...Ch. 6.2 - Prob. 44ECh. 6.2 - Prob. 45ECh. 6.2 - ,There are 51 houses on a street, Each house has...Ch. 6.2 - Letibe an irrational number, Showthatfor some...Ch. 6.2 - Prob. 48ECh. 6.2 - Prob. 49ECh. 6.3 - i. List all the permutations of{a, b,c}.Ch. 6.3 - How many different permutations are there of the...Ch. 6.3 - How many permutations of{a, b,c, d,e.fg]end withCh. 6.3 - LetS = {i,2, 3,4, 5}. List all the 3-permutations...Ch. 6.3 - Find the value of each of these quantities P(6,3)...Ch. 6.3 - Find the value of each of these quantities. CCs,i)...Ch. 6.3 - Find the number of 5-permutations of a set Kith...Ch. 6.3 - In how many different orders can five runners...Ch. 6.3 - Prob. 9ECh. 6.3 - There are six different candidates for governor of...Ch. 6.3 - ii.How many bit strings of length 10 contain...Ch. 6.3 - IE.How many bit strings of length12contain exactly...Ch. 6.3 - A group contains n men and n women. How many ways...Ch. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - Prob. 16ECh. 6.3 - Prob. 17ECh. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Prob. 20ECh. 6.3 - Prob. 21ECh. 6.3 - Prob. 22ECh. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Prob. 27ECh. 6.3 - Prob. 28ECh. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 32ECh. 6.3 - Prob. 33ECh. 6.3 - Prob. 34ECh. 6.3 - Prob. 35ECh. 6.3 - Prob. 36ECh. 6.3 - Prob. 37ECh. 6.3 - Prob. 38ECh. 6.3 - Prob. 39ECh. 6.3 - Prob. 40ECh. 6.3 - Prob. 41ECh. 6.3 - Find the number of circular 3-permutations...Ch. 6.3 - Prob. 43ECh. 6.3 - Prob. 44ECh. 6.3 - How many ways are there for a horse race with...Ch. 6.4 - Find the expansion of (r + using combinatorial...Ch. 6.4 - Find the expansion of Cr + j,)5 using...Ch. 6.4 - Find the expansionCh. 6.4 - Find the coefficient of in Cr + y)13.Ch. 6.4 - How many terms are therein the expansion of...Ch. 6.4 - What isthecoefficient of .v in (1 +1)Ch. 6.4 - What is the coefficient of i9 in (2 - 1)Ch. 6.4 - What is the coefficient ofxsy9 in the expansion of...Ch. 6.4 - What is the coefficient of xloly" in the expansion...Ch. 6.4 - Prob. 10ECh. 6.4 - Prob. 11ECh. 6.4 - IS. Use the binomial theorem to find the...Ch. 6.4 - *3-Use the binomial theorem to find the...Ch. 6.4 - Give a formula for the coefficient ofi^in the...Ch. 6.4 - Prob. 15ECh. 6.4 - The row of Pascal’s triangle containing the...Ch. 6.4 - What is the r ow of Pascal's triangle containing...Ch. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - so. Use Exercise 18 andCorollary 1to show that...Ch. 6.4 - Prob. 21ECh. 6.4 - Suppose thatbis an integer withb> 7. Use the...Ch. 6.4 - Prove Pas cal’s identity, u sing the formula for...Ch. 6.4 - Suppose that t andnare integers withi which...Ch. 6.4 - Provethatifnandfcareintegers^th i< fc using a...Ch. 6.4 - Prove the identity (")(') = (J)(Xf), whenever n,...Ch. 6.4 - Prob. 27ECh. 6.4 - Prob. 28ECh. 6.4 - Letnbe a positive integer. Show thatCh. 6.4 - Prob. 30ECh. 6.4 - Prove the hockey-stick identity ('?’)...Ch. 6.4 - Show that if ra is a positive integer, then =2t" i...Ch. 6.4 - Prob. 33ECh. 6.4 - Prob. 34ECh. 6.4 - Prob. 35ECh. 6.4 - Prove the binomial theorem using mathematical...Ch. 6.4 - In this exercise we will count the number of paths...Ch. 6.4 - Prob. 38ECh. 6.4 - Prob. 39ECh. 6.4 - Prob. 40ECh. 6.4 - Prob. 41ECh. 6.4 - Prob. 42ECh. 6.4 - Determine a formula involving binomial...Ch. 6.5 - In how many different wavs can five elements be...Ch. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - How many different ways are there to choose a...Ch. 6.5 - A bagel shop has onion bagels, poppy seed bagels,...Ch. 6.5 - io. A croissant shop has plain croissants, cherry...Ch. 6.5 - ii. Howmany ways are there to choose eight coins...Ch. 6.5 - Homy different combinations of pennies, nickels,...Ch. 6.5 - Prob. 13ECh. 6.5 - How many solutions are there to the equation -T| +...Ch. 6.5 - How many solutions are there to the equation -T |...Ch. 6.5 - i6. How many solutions are there to the equation...Ch. 6.5 - strings of 10 ternary digits (o, 1. or 2) are...Ch. 6.5 - ,How many strings of 20-decima] digits are there...Ch. 6.5 - Prob. 19ECh. 6.5 - How many solutions are there to the inequality .ii...Ch. 6.5 - i. A Swedish tour guide has devised a clever way...Ch. 6.5 - w many ways can an airplane pilot be scheduled for...Ch. 6.5 - How many ways are there to distribute six...Ch. 6.5 - How many ways are there to distribute 12...Ch. 6.5 - Howmany wavs aiethereto distribute 12...Ch. 6.5 - Prob. 26ECh. 6.5 - How many positive integers less than 1,000,000...Ch. 6.5 - a8. How many positive integers less than 1,000,000...Ch. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Prob. 31ECh. 6.5 - How many different strings can be made from the...Ch. 6.5 - How many different strings can be made from the...Ch. 6.5 - different strings can be made from the letters...Ch. 6.5 - How many different strings can be made from the...Ch. 6.5 - How many strings idth five or more characters can...Ch. 6.5 - How many strings with seven or more characters can...Ch. 6.5 - How many different bit strings can be formed using...Ch. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - i....Ch. 6.5 - Prob. 42ECh. 6.5 - How many ways are. there to deal hands of seven...Ch. 6.5 - In bridge. the 52 cards of a standard deck are...Ch. 6.5 - How many ways are there to deal hands of five...Ch. 6.5 - , In how many ways can a dozen books be placed on...Ch. 6.5 - How many ways cannbooks be placed on t...Ch. 6.5 - Prob. 48ECh. 6.5 - Prob. 49ECh. 6.5 - Prob. 50ECh. 6.5 - Prob. 51ECh. 6.5 - How many ways are there to distribute five...Ch. 6.5 - Prob. 53ECh. 6.5 - 54-How many ways are there to put five temporary...Ch. 6.5 - Prob. 55ECh. 6.5 - Prob. 56ECh. 6.5 - Prob. 57ECh. 6.5 - 8,Howmany ways are thereto pack eightidentical...Ch. 6.5 - Prob. 59ECh. 6.5 - 6o. How many ways are there to distribute five...Ch. 6.5 - 6i. How many ways are there to distribute five...Ch. 6.5 - Suppose that a basketball league has 32 teams,...Ch. 6.5 - f 63. Suppose that a weapons inspector must...Ch. 6.5 - Howmanv dififerentterms are therein the expansion...Ch. 6.5 - Prob. 65ECh. 6.5 - Prob. 66ECh. 6.5 - Find the coefficient ofi3y2z5 in Qc + y + z)Ch. 6.5 - How many terms are there in the expansionCh. 6.6 - ...Ch. 6.6 - ...Ch. 6.6 - Prob. 3ECh. 6.6 - Prob. 4ECh. 6.6 - Find the next larger permutation in lexicographic...Ch. 6.6 - Find the next larger permutation in lexicographic,...Ch. 6.6 - Use Algorithm 1 to generate the 24 permutations of...Ch. 6.6 - Prob. 8ECh. 6.6 - Use Algorithm 3 to listallthe 3-combinations of{1,...Ch. 6.6 - Show that Algorithm1produces the next larger...Ch. 6.6 - Show that Algorithm 3 produces the next larger...Ch. 6.6 - Develop an algorithm for generating the...Ch. 6.6 - List all 3-permutations of {1,2,3,4,5}. The...Ch. 6.6 - Find the Cantor digits an ti2,that correspond to...Ch. 6.6 - Prob. 15ECh. 6.6 - i6,Find the permutations of {1,2,3,4,5} that...Ch. 6.6 - Prob. 17ECh. 6 - Explain how the sum and product rules can be used...Ch. 6 - Explain how to find the number of bit strings of...Ch. 6 - Prob. 3RQCh. 6 - How can yon find the number of possible outcomes...Ch. 6 - How can you find the number of bit strings...Ch. 6 - State the pigeonhole principle, Explain how the...Ch. 6 - State the generalized pigeonhole principle....Ch. 6 - ft What is the difference between an r-combination...Ch. 6 - What i s Pas cal's tri angle? How can arow of...Ch. 6 - What is meant by a combinatorial proof of an...Ch. 6 - ii. Explain how to prove Pascal's identity using a...Ch. 6 - Stateth e bin omial th eor em. Explain how to pr o...Ch. 6 - Explain how to find a formula for the number of...Ch. 6 - Letnand r be positive integers. Explain why the...Ch. 6 - Prob. 15RQCh. 6 - Prob. 16RQCh. 6 - a) How many ways are there to deal hands of five...Ch. 6 - Describe an algorithm for generating all the...Ch. 6 - i. How many ways are there to choose 6 items from...Ch. 6 - a.H 01 v many ways ar e ther e to ch o o se1o...Ch. 6 - Prob. 3SECh. 6 - How many strings of length10either start with ooo...Ch. 6 - Prob. 5SECh. 6 - Prob. 6SECh. 6 - Prob. 7SECh. 6 - Hoi v many positive integers less than iqoo have...Ch. 6 - Prob. 9SECh. 6 - Prob. 10SECh. 6 - Prob. 11SECh. 6 - How many people are needed to guarantee that at...Ch. 6 - Show that given anv set of 10 positive integers...Ch. 6 - Prob. 14SECh. 6 - Prob. 15SECh. 6 - Prob. 16SECh. 6 - Show that in a sequence ofmintegers there exists...Ch. 6 - Prob. 18SECh. 6 - Show that the decimal expansion of a rational...Ch. 6 - Once a computer worm infects a personal computer...Ch. 6 - si.How many ways are there to choose a dozen...Ch. 6 - ss.Findn if P(n,2] = 110. J\?i, n] = 5040....Ch. 6 - Prob. 23SECh. 6 - Show that ifnandrare nonnegative integers and n >...Ch. 6 - Prob. 25SECh. 6 - Give a combinatorial proof ofCorollary 2ofSection...Ch. 6 - Prob. 27SECh. 6 - a8. Prove using mathematical induction that O>• 2)...Ch. 6 - Prob. 29SECh. 6 - Show that V7' XIt. I = (’) if nis an integer withCh. 6 - Prob. 31SECh. 6 - Prob. 32SECh. 6 - How many bit strings of length n, where n > 4,...Ch. 6 - Prob. 34SECh. 6 - Prob. 35SECh. 6 - Prob. 36SECh. 6 - How many ways are there to assign 24 students to...Ch. 6 - Prob. 38SECh. 6 - - How many solutions are there to the equation xt...Ch. 6 - How many different strings can be made from the...Ch. 6 - How many subsets of a set with ten el e m ents...Ch. 6 - Prob. 42SECh. 6 - Prob. 43SECh. 6 - How many ways are. there to seat six boys and...Ch. 6 - How many ways are there to distribute six objects...Ch. 6 - How many ways are there to distribute five obj...Ch. 6 - Find these signless Stirling numb er s of the...Ch. 6 - Show that ifnis a positive integer, then ,Ch. 6 - Prob. 49SECh. 6 - Prob. 50SECh. 6 - Prob. 51SECh. 6 - j2, How many n-element RXA sequences consist of 4...Ch. 6 - *53. Suppose that when an enzyme that breaks RXA...Ch. 6 - Suppose that when an enzyme that breaks RXA chains...Ch. 6 - Devise an algorithm for generating all the...Ch. 6 - Devise an algorithm for generating all the...Ch. 6 - Prob. 57SECh. 6 - Prob. 58SECh. 6 - Prob. 1CPCh. 6 - Prob. 2CPCh. 6 - Prob. 3CPCh. 6 - Prob. 4CPCh. 6 - Given a positive integern,listallthe permutations...Ch. 6 - Prob. 6CPCh. 6 - Prob. 7CPCh. 6 - Prob. 8CPCh. 6 - Prob. 9CPCh. 6 - Prob. 10CPCh. 6 - Prob. 1CAECh. 6 - Prob. 2CAECh. 6 - Prob. 3CAECh. 6 - Prob. 4CAECh. 6 - Prob. 5CAECh. 6 - Prob. 6CAECh. 6 - Prob. 7CAECh. 6 - Prob. 8CAECh. 6 - Prob. 9CAECh. 6 - Describe some of the earliest uses of the...Ch. 6 - Prob. 2WPCh. 6 - Discuss the importance of combinatorial reasoning...Ch. 6 - Jlanv combinatonal identities are described in...Ch. 6 - Prob. 5WPCh. 6 - Prob. 6WPCh. 6 - Prob. 7WPCh. 6 - Describe the latest discoveries of values and...Ch. 6 - Prob. 9WPCh. 6 - Prob. 10WP

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