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4. Permutations Permutations: Notation; n Objects Taken k at a Time Permutations of Nondistinguishable Objects Combinations Combinatorics Sequences, Series, and Combinatorics > Combinatorics Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1), or with colored balls, as in. The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×2×1, which number is called "n factorial" and written "n! ". Permutations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/combinatorics-57/permutations-238- 5523?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess One stack of cards in a game of solitaire View on Boundless.com Sequences, Series, and Combinatorics > Combinatorics

6. When deciding permutations of a subset from a larger set, it is often useful to divide one factorial by another to determine the number of permutations possible. For example, the first six cards from a deck of cards would have [Equation 1] permutations possible, or about 14.7 billion. Sequences, Series, and Combinatorics > Combinatorics Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/combinatorics-57/permutations-238- 5523?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 1 View on Boundless.com

7. If all objects in consideration are distinct, they can be arranged in n! permutations, where n represents the number of objects. If not all the objects in a set of n unique elements are chosen, the above formula can be modified to: where k represents the number of selected elements. When solving for quotients of factorials, the terms of the denominator can cancel with the terms of the numerator, thus eliminating perhaps the majority of terms to be multiplied. Permutations: Notation; n Objects Taken k at a Time Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/combinatorics-57/permutations- notation-n-objects-taken-k-at-a-time-239- 10905?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Showing the 6 permutations of 3 colored balls (red, green and blue) View on Boundless.com Sequences, Series, and Combinatorics > Combinatorics

8. Some sets include repetitions of certain elements. In these cases, the number of possible permutations of the items cannot be expressed by n!, where n represents the number of elements, because this calculation would include a multiplicity of possible states. To correct for the "multiplicity" of certain permutations, divide the factorial of the total number of elements by the product of the factorials of the number of each repeated element. Permutations of Nondistinguishable Objects Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/combinatorics-57/permutations-of- nondistinguishable-objects-240- 10908?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless View on Boundless.com Sequences, Series, and Combinatorics > Combinatorics

9. The expression for number of permutations with repeated elements is: [Equation 2] where n is the total number of terms in a sequence n1, n2, and n3 are the number of repetitions of different elements. Sequences, Series, and Combinatorics > Combinatorics Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/combinatorics-57/permutations-of- nondistinguishable-objects-240- 10908?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Equation 2 View on Boundless.com

10. A combination is a mathematical concept where one counts the number of ways one can select several elements out of a larger group. Unlike a permutation, when determining the number of combinations, order does not matter. Combinations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/combinatorics-57/combinations-241- 5867?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Pascal's triangle View on Boundless.com Sequences, Series, and Combinatorics > Combinatorics

11. Formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient: [Equation 3], which can be written using factorials as [Equation 4] whenever [Equation 5] and which is zero when k>n. Sequences, Series, and Combinatorics > Combinatorics Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/combinatorics-57/combinations-241- 5867?campaign_content=book_196_section_57&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 3 View on Boundless.com Equation 4 View on Boundless.com Equation 5 View on Boundless.com

12. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

13. Key terms binomial coefficient A coefficient of any of the terms in the expansion of the binomial (x+y)n. factorial The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 * 2 * 3 * 4 * 5 = 120. multiplicity the number of values for which a given condition holds permutation An ordering of a finite set of distinct elements. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics

14. Showing the 6 permutations of 3 colored balls (red, green and blue) If one has three different colored balls, there are six distinct permutations (or ways to organize them) into a different order, as shown. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Permutations RGB." CC BY-SA http://en.wikipedia.org/wiki/File:Permutations_RGB.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Permutations_RGB.svgView on Boundless.com Sequences, Series, and Combinatorics

15. The 6 permutation of 3 balls If one has three different colored balls, there are six distinct ways to organize them into a different order, as shown. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Permutations RGB." CC BY-SA http://en.wikipedia.org/wiki/File:Permutations_RGB.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Permutations_RGB.svgView on Boundless.com Sequences, Series, and Combinatorics

16. One stack of cards in a game of solitaire To find out how many possible combinations of cards there are below the seven, we use the concept of permutations to calculate the possible arrangements of cards. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Permutations." CC BY 3.0 http://cnx.org/content/m19072/latest/?collection=col10624 View on Boundless.comCC BY 3.0http://cnx.org/content/m19072/latest/?collection=col10624View on Boundless.com Sequences, Series, and Combinatorics

17. Pascal's triangle The binomial coefficient can be arranged to form Pascal's trangle. The first row is the terms for (x+y)^0, the second for (x+y)^1, the third for (x=y)^2, which is 1x^2+2xy+y^2. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Pascal's triangle 5." CC BY-SA http://en.wikipedia.org/wiki/File:Pascal%2527s_triangle_5.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Pascal%2527s_triangle_5.svgView on Boundless.com Sequences, Series, and Combinatorics

18. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics How many different ways are there to arrange the letters in the word QUESTION? A) 5,040 ways B) 720 ways C) 40,320 ways D) 120 ways

19. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics How many different ways are there to arrange the letters in the word QUESTION? A) 5,040 ways B) 720 ways C) 40,320 ways D) 120 ways

20. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics How many 5-letter strings of letters from the alphabet can be formed if letters cannot be repeated? A) 15,600 B) 165,765,600 C) 7,893,600 D) 358,800

21. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics How many 5-letter strings of letters from the alphabet can be formed if letters cannot be repeated? A) 15,600 B) 165,765,600 C) 7,893,600 D) 358,800

22. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics How many different ways are there to choose a president, vice president, secretary, and treasurer from a group of 50 people? A) 230,300 B) 5,527,200 C) 117,600 D) 254,251,200

23. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics How many different ways are there to choose a president, vice president, secretary, and treasurer from a group of 50 people? A) 230,300 B) 5,527,200 C) 117,600 D) 254,251,200

24. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics How many different ways are there to make a six-digit license plate number (no letters) if the numbers cannot be repeated? A) 1,000,000 B) 30,240 C) 604,800 D) 151,200

25. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics How many different ways are there to make a six-digit license plate number (no letters) if the numbers cannot be repeated? A) 1,000,000 B) 30,240 C) 604,800 D) 151,200

26. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics How many different ways can you rearrange the letters in the word BANANA? A) 120 B) 360 C) 720 D) 60

27. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics How many different ways can you rearrange the letters in the word BANANA? A) 120 B) 360 C) 720 D) 60

28. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics There are 12 marbles: 1 green, 2, blue, 2 orange, 3 red, and 4 yellow. How many different ways are there to arrange these marbles in a straight line (assuming that same colored marbles are identical)? A) 166,320 B) 1,663,200 C) 479,001,600 D) 831,600

29. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics There are 12 marbles: 1 green, 2, blue, 2 orange, 3 red, and 4 yellow. How many different ways are there to arrange these marbles in a straight line (assuming that same colored marbles are identical)? A) 166,320 B) 1,663,200 C) 479,001,600 D) 831,600

30. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics How many different 3-person committees can be created from a group of 45 people? A) 14,190 B) 85,140 C) 25,620 D) 20,650

31. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics How many different 3-person committees can be created from a group of 45 people? A) 14,190 B) 85,140 C) 25,620 D) 20,650

32. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics You have 30 songs but can only fit 25 on a CD. How many different combinations of songs can you put on the CD? A) 23,751 B) 142,506 C) 150,540 D) 275

33. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics You have 30 songs but can only fit 25 on a CD. How many different combinations of songs can you put on the CD? A) 23,751 B) 142,506 C) 150,540 D) 275

34. Attribution Wikipedia. "Permutation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/PermutationCC BY-SA 3.0http://en.wikipedia.org/wiki/Permutation Connexions. "Permutations." CC BY 3.0 http://cnx.org/content/m19072/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m19072/latest/?collection=col10624 Wiktionary. "permutation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/permutationCC BY-SA 3.0http://en.wiktionary.org/wiki/permutation Wiktionary. "factorial." CC BY-SA 3.0 http://en.wiktionary.org/wiki/factorialCC BY-SA 3.0http://en.wiktionary.org/wiki/factorial Wikipedia. "Combination." CC BY-SA 3.0 http://en.wikipedia.org/wiki/CombinationCC BY-SA 3.0http://en.wikipedia.org/wiki/Combination Connexions. "Combinations." CC BY 3.0 http://cnx.org/content/m19071/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m19071/latest/?collection=col10624 Wiktionary. "binomial coefficient." CC BY-SA 3.0 http://en.wiktionary.org/wiki/binomial+coefficientCC BY-SA 3.0http://en.wiktionary.org/wiki/binomial+coefficient Wikipedia. "Permutation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/PermutationCC BY-SA 3.0http://en.wikipedia.org/wiki/Permutation Wiktionary. "permutation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/permutationCC BY-SA 3.0http://en.wiktionary.org/wiki/permutation Wiktionary. "factorial." CC BY-SA 3.0 http://en.wiktionary.org/wiki/factorialCC BY-SA 3.0http://en.wiktionary.org/wiki/factorial Wikipedia. "Factorial." CC BY-SA 3.0 http://en.wikipedia.org/wiki/FactorialCC BY-SA 3.0http://en.wikipedia.org/wiki/Factorial Wiktionary. "multiplicity." CC BY-SA 3.0 http://en.wiktionary.org/wiki/multiplicityCC BY-SA 3.0http://en.wiktionary.org/wiki/multiplicity Wiktionary. "permutation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/permutationCC BY-SA 3.0http://en.wiktionary.org/wiki/permutation Wikipedia. "Permutation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/PermutationCC BY-SA 3.0http://en.wikipedia.org/wiki/Permutation Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics