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8-7 Pascal’s Triangle

Pascal's Triangle is a mathematical pattern that helps calculate combinations. This text explains the properties of Pascal's Triangle, how to use it, and provides examples.

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8-7 Pascal’s Triangle

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  1. 8-7 8-7 Pascal’s Triangle

  2. 8-7 Pascal’s Triangle For any given value of n, there are n+1 different calculations of combinations: 0C0 1C0 =1 n = 0, 1 calculation n = 1, 2 calculations n = 2, 3 calculations etc… 1C1 =1 =1 2C0 2C1 2C2 =1 =2 =1 3C0 =1 3C1 3C2 3C3 =3 =3 =1 4C0 =1 4C1 4C2 4C3 4C4 =4 =6 =4 =1

  3. 8-7 Pascal’s Triangle For any given value of n, there are n+1 different calculations of combinations: 0C0 = 1C0 = 1 1C1 = 1 1 2C0 = 2C1 = 2C2 = 1 2 1 3C0 = 1 3C1 = 3C2 = 3C3 = 3 3 1 4C0 = 4C1 = 4C2 = 4C3 = 4 4C4 = 1 4 6 1

  4. 8-7 Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 4 1 4 6 1

  5. Row 0 1 • Row 1 1 1 • Row 2 1 2 1 • Row 3 1 3 3 1 • Row 4 1 4 6 4 1 • Row 5 1 5 10 10 5 1 • Row 6 1 6 15 20 15 6 1 • Row 7? • See any patterns?

  6. 8-7 Properties of Pascal’s • The first and the last term are always 1. • The second and next to last terms in the nth row are n • Each row is symmetric • The sum of the terms in row n is: 2n 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

  7. 8-7 The number of the row always starts with 0 The number of the terms in a row is always one greater than the row number 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Pascal’s Triangle R n: R 0: R 1: R 2: R 3: R 4: Recursively, every term can be found by finding the sum of the two terms diagonally above it. Explicitly, every term in any row can be found using the combinations formula: Where n is the number of the row, and r+1 is the number of the term

  8. 8-7 Using Pascal’s Triangle What is 4C2? What is the 6th term of row 50 of pascal’s triangle? 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Combinations/Pascal’s

  9. Examples: • Find the first 4 terms in row 9 of Pascal’s Triangle • Construct row 12 if the first 6 terms in row 11 are: 1,11,55,165,330,462

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