PASCAL’S TRIANGLE Todd Cochrane 1 / 29 Robert’s Dream In his dream, Robert and the Number Devil build a giant pyramid of blocks. Being two-dimensional they agree to call it a triangle of blocks. Then they start marking the blocks with a felt pen, labeling the top block 1, the next two 1,1, and then each block thereafter the sum of the two values above it. Thus, the third row is 1,2,1, the fourth 1,3,3,1 and so on. 2 / 29 Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Rule: Each term in Pascal’s triangle is the sum of the two terms above it. Pascal’s triangle is named after Blaise Pascal, who put together many of its properties in 1653. 3 / 29 Triangle Numbers. Diagonal 3: 1; 3; 6; 10; 15; 21;::: Diagonal 4: 1; 4; 10; 20; 35; 56;::: Tetrahedral Numbers. Diagonal’s of Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Diagonals: Note, the Southwesterly and Southeasterly diagonals are the same. Diagonal 1: 1; 1; 1; 1;::: (yawn) Diagonal 2: 1; 2; 3; 4; 5; : : : ::: Natural Numbers . 4 / 29 . Triangle Numbers. Diagonal 4: 1; 4; 10; 20; 35; 56;::: Tetrahedral Numbers. Diagonal’s of Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Diagonals: Note, the Southwesterly and Southeasterly diagonals are the same. Diagonal 1: 1; 1; 1; 1;::: (yawn) Diagonal 2: 1; 2; 3; 4; 5; : : : ::: Natural Numbers Diagonal 3: 1; 3; 6; 10; 15; 21;::: . 5 / 29 . Diagonal 4: 1; 4; 10; 20; 35; 56;::: Tetrahedral Numbers. Diagonal’s of Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Diagonals: Note, the Southwesterly and Southeasterly diagonals are the same. Diagonal 1: 1; 1; 1; 1;::: (yawn) Diagonal 2: 1; 2; 3; 4; 5; : : : ::: Natural Numbers Diagonal 3: 1; 3; 6; 10; 15; 21;::: Triangle Numbers. 6 / 29 . Diagonal’s of Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Diagonals: Note, the Southwesterly and Southeasterly diagonals are the same. Diagonal 1: 1; 1; 1; 1;::: (yawn) Diagonal 2: 1; 2; 3; 4; 5; : : : ::: Natural Numbers Diagonal 3: 1; 3; 6; 10; 15; 21;::: Triangle Numbers. Diagonal 4: 1; 4; 10; 20; 35; 56;::: Tetrahedral Numbers. 7 / 29 . Triangle Numbers T1 = 1 T2 = 1 + 2 = 3 T3 = 1 + 2 + 3 = 6 T4 = 1 + 2 + 3 + 4 = 10 The n-th triangle number is the sum of the first n natural numbers 1 1 1 1 2 1 1 3 3 1 ; 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 8 / 29 Tetrahedral Numbers Let Hn denote the n-th tetrahedral number. H1 = 1, H2 = 4, H3 = 10,... There are two ways to define Hn. Algebraic: The n-th tetrahedral number Hn is the sum of the first n triangle numbers. Geometric: The n-th tetrahedral number Hn is the number of balls required to form a tetrahedral stack with n balls along each edge. 9 / 29 Fourth Tetrahedral Number 10 / 29 • A Pentatopic Number is obtained by adding Tetrahedral numbers. 1 + 4 = 5 1 + 4 + 10 = 15 1 + 4 + 10 + 20 = 35 Sum of the Diagonal Numbers in Pascal’s Triangle Natural Numbers: 1,2,3,4,5,... Triangle Numbers: 1,3,6,10,15,... Tetrahedral Numbers: 1,4,10,20,35,... • A Triangle Number is obtained by adding Natural numbers. • A Tetrahedral Number is obtained by adding Triangle numbers. What comes next? 11 / 29 Sum of the Diagonal Numbers in Pascal’s Triangle Natural Numbers: 1,2,3,4,5,... Triangle Numbers: 1,3,6,10,15,... Tetrahedral Numbers: 1,4,10,20,35,... • A Triangle Number is obtained by adding Natural numbers. • A Tetrahedral Number is obtained by adding Triangle numbers. What comes next? • A Pentatopic Number is obtained by adding Tetrahedral numbers. 1 + 4 = 5 1 + 4 + 10 = 15 1 + 4 + 10 + 20 = 35 12 / 29 The Hockey Stick Identity: Rule: The sum of the numbers along a diagonal of Pascal’s Triangle equals the number in the next row shifted one place to the right (for Southwesterly diagonals). 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Example: 1 + 4 + 10 + 20 = 35 13 / 29 Rule: The coefficients in the binomial expansion of (x + y)n are the numbers in the n-th row of Pascal’s Triangle. Lets see if we can understand this. (x + y)2 = (x + y)(x + y) = x2 + yx + xy + y 2; FOIL, Distributive Law = x2 + 2xy + y 2; Commutative law Binomial Expansions: (x + y)0 = 1 (x + y)1 = 1x + 1y (x + y)2 = 1x2 + 2xy + 1y 2 (x + y)3 = 1x3 + 3x2y + 3xy 2 + 1y 3 (x + y)4 = 1x4 + 4x3y + 6x2y 2 + 4xy 3 + 1y 4 14 / 29 Binomial Expansions: (x + y)0 = 1 (x + y)1 = 1x + 1y (x + y)2 = 1x2 + 2xy + 1y 2 (x + y)3 = 1x3 + 3x2y + 3xy 2 + 1y 3 (x + y)4 = 1x4 + 4x3y + 6x2y 2 + 4xy 3 + 1y 4 Rule: The coefficients in the binomial expansion of (x + y)n are the numbers in the n-th row of Pascal’s Triangle. Lets see if we can understand this. (x + y)2 = (x + y)(x + y) = x2 + yx + xy + y 2; FOIL, Distributive Law = x2 + 2xy + y 2; Commutative law 15 / 29 Calculating (x + y)3 and (x + y)4 Lets calculate (x + y)3, given (x + y)2: (x + y)3 = (x + y)2(x + y) = (x2 + 2xy + y 2)(x + y) = (x2 + 2xy + y 2)x + (x2 + 2xy + y 2)y Once more: Lets calculate (x + y)4, given (x + y)3: (x + y)4 = (x + y)3(x + y) = (x3 + 3x2y + 3xy 2 + y 3)(x + y) = (x3 + 3x2y + 3xy 2 + y 3)x + (x3 + 3x2y + 3xy 2 + y 3)y 16 / 29 Using Pascal’s Triangle to calculate (x + y)6 Example: Use Pascals Triangle to find (x + y)6. 1 1 1 1 2 1 1 3 3 1 ; 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 17 / 29 The Magic Number Eleven Calculate 11; 112; 113, and find the pattern. 18 / 29 The Magic Number Eleven Calculate 11; 112; 113, and find the pattern. 19 / 29 Sum of the Numbers in a Row of Pascal’s Triangle. Row Sum Total R0 1 1 1 R1 1 + 1 2 1 1 R2 1 + 2 + 1 4 1 2 1 , R3 1 3 3 1 R4 1 4 6 4 1 R5 1 5 10 10 5 1 R6 1 6 15 20 15 6 1 . Rule: The sum of the numbers in the n-th row of Pascal’s Triangle is WHY? 20 / 29 Rule: The sums of the numbers on the shallow diagonals are the The Shallow Diagonals of Pascal’s Triangle A shallow diagonal is formed by moving left one unit and down a diagonal one unit to get to the next number. 1 Sum Total 1 1 1 1 1 2 1 1 + 1 2 1 3 3 1 2 + 1 ; 1 4 6 4 1 1 + 3 + 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 21 / 29 The Shallow Diagonals of Pascal’s Triangle A shallow diagonal is formed by moving left one unit and down a diagonal one unit to get to the next number. 1 Sum Total 1 1 1 1 1 2 1 1 + 1 2 1 3 3 1 2 + 1 ; 1 4 6 4 1 1 + 3 + 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Rule: The sums of the numbers on the shallow diagonals are the 22 / 29 Rule: The sum of the squares of the numbers in n-th row Rn is the number in the middle of row R2n. Sum of squares of numbers in a row.