8 7 11272020 8 7 Pascals Triangle 8

8 -7 11/27/2020 8 -7 Pascal’s Triangle

8 -7 Pascal’s Triangle 11/27/2020 For any given value of n, there are n+1 different calculations of combinations: 0 C 0 =1 n = 0, 1 calculation 1 C 1 =1 n = 1, 2 calculations 1 C 0 =1 2 C 1 =2 2 C 2 =1 n = 2, 3 calculations C =1 C =3 C =1 etc… 3 0 4 C 0 3 =1 1 4 C 1 3 =4 2 4 C 2 3 =6 3 4 C 3 =4 4 C 4 =1

8 -7 Pascal’s Triangle 11/27/2020 For any given value of n, there are n+1 different calculations of combinations: 0 C 0 = 1 C = 1 1 C 0 = 1 1 1 2 C 0 = 1 2 C 1 = 2 2 C 2 = 1 1 3 C 1 = 3 3 C 2 = 3 3 C 3 = 1 3 C 0 = 4 C 0 = 1 4 C 1 = 4 4 C 2 = 6 4 C 3 = 4 4 C 4 = 1

8 -7 Pascal’s Triangle 11/27/2020 1 1 1 2 1 3 3 1 1 4 6 4 1

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

8 -7 11/27/2020 Properties of Pascal’s 1. The first and the last term are always 1. 2. The second and next to last terms in the nth row are n 3. Each row is symmetric 4. The sum of the terms in row n is: 1 1 1 2 n 1 1 1 2 3 4 1 3 6 1 4 1

8 -7 11/27/2020 Pascal’s Triangle • 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 R n: R 0: R 1: 1 R 2: 1 R 3: 4 R 4: 1 1 2 3 1 3 6 1 4 1 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 -7 11/27/2020 Combinations/Pascal’s Using Pascal’s Triangle What is 4 C 2? 1 1 1 What is the 6 th term of row 50 of pascal’s triangle? 1 1 1 2 3 4 1 3 6 1 4 1

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