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Permutations Combinations Pascal’s triangle Binomial Theorem

Permutations Combinations Pascal’s triangle Binomial Theorem. Permutations. 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:

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Permutations Combinations Pascal’s triangle Binomial Theorem

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  1. Permutations Combinations Pascal’s triangle Binomial Theorem

  2. Permutations • 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 P = 3 2  1 = 6 3 3

  3. Permutations • 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 P = 3 2 = 6 3 2 P = 3 3 1

  4. Permutations How many arrangements of five letters can be made from the letters in the word FIRST? F 5 I 4 R 3 2 S T 1     P = 5 4 3 2 1 = 120 5 5

  5. Combinations • 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 3 C = 1 3

  6. Combinations • 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 3 C = 3 2 3 C = 3 1

  7. 10  9 2  1 ––––– = Combinations Fred has a voucher to pick any two of the top 10 PS3 games! How many different combinations of 2 games can he pick? 10 C = 45 2

  8. Pascal’s Triangle 1 1 1 1 2 1 1 3 31 1 4 6 4 1 1 5 10 10 5 1

  9. 1 1 1 2 1 Binomial Theorem 1 (x+y)1=x+y (x+y)2=x2+ 2xy + y2 1 3 31 (x+y)3=x3+ 3x2y1 + 3x1y2+y3 1 4 6 4 1 (x+y)4=x4+ 4x3y1+ 6x2y2 + 4x1y3+y4

  10. Binomial Theorem (x+y)6 6C0 x6 x6+ 6x5y1+ 15x4y2 + 20x3y3+ 15x2y4 + 6x1y5 +y6 6C1 x5y1 + 6C2 15 x4y2 + 6C3 20 x3y3 + 6C4 15 x2y4 + 6C5 6 x1y5 + 6 6C6 y6 +

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