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Permutations

Combinations

Pascal’s triangle

Binomial Theorem

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• These are arrangements in which the order matters.• Consider three letters a, b, c.• How many arrangements of these three letters can be

made using each once? • There are six possible arrangements of three letters:

abc acb bac bca cab cba = 6 permutations

PermutationsPermutations

P3 3 = 3 2 1 = 6

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• How many arrangements of two letters can be made from three letters?

ab ac ba bc ca cb = 6 permutations

• How many arrangements of one letter can be made from three letters?

a b c = 3 permutations

PermutationsPermutations

P3 2 = 3 2 = 6

P3 1 = 3

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F SI TR

PermutationsPermutationsHow many arrangements of five letters can be made from the letters in the word FIRST?

5 24 13P5 5 = 5 4 3 2 1 =

120

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CombinationsCombinations• These are groups of things where order does not

matter.

• Consider three letters a, b, c. How many combinations of three letters can be made taking each once?

• There is only 1, abc = 1 combination

C3

3= 1

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CombinationsCombinations• How many combinations of two letters can be made

from three letters?

ab, ac, bc = 3 combinations

• How many combinations of one letter can be made from three letters?

a, b, c = 3 combinations

C3

2= 3

C3

1= 3

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CombinationsCombinationsFred has a voucher to pick any two of the top 10

PS3 games! How many different combinations of

2 games can he pick?

C10

2= 45

10 92 1–––––=

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1

11

11 2

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Pascal’s TrianglePascal’s Triangle

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Binomial TheoremBinomial Theorem

1

11

11 2

1 3 3 1

1 4 6 4 1

(x + y)1 = x + y

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y1 + 3x1y2 + y3

(x + y)4 = x4 + 4x3y1 + 6x2y2 + 4x1y3 + y4

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x5y16

Binomial TheoremBinomial Theorem

(x + y)6

6C0 x6 6C16C2 x4y2+ + 6C3 x3y3+ 6C4 x2y4+ 6C5 x1y5+ 6C6 y6+15 20 15 6x6 + 6x5y1 + 15x4y2 + 20x3y3 + 15x2y4 + 6x1y5 + y6

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