PASCAL’S TRIANGLE

PASCAL’S TRIANGLE

* ABOUT THE MAN

* CONSTRUCTING THE TRIANGLE

* PATTERNS IN THE TRIANGLE

* PROBABILITY AND THE TRIANGLE

Blaise PascalJUNE 19,1623-AUGUST 19, 1662

*French religious philosopher, physicist, and mathematician.*“Thoughts on Religion”. (1655)*Syringe, and Pascal’s Law. (1647-1654)*First Digital Calculator. (1642-1644) *Modern Theory of Probability/Pierre de Fermat. (1654)*Chinese mathematician Yanghui, 500 years before Pascal; Eleventh century Persian mathematician and poet Omar Khayam.*Pascal was first to discover the importance of the patterns.

CONSTRUCTING THE TRIANGLE

* START AT THE TOP OF THE TRIANGLE WITH

THE NUMBER 1; THIS IS THE ZERO ROW.

* NEXT, INSERT TWO 1s. THIS IS ROW 1.

* TO CONSTRUCT EACH ENTRY ON THE NEXT ROW, INSERT 1s ON EACH END,THEN ADD THE TWO ENTRIES ABOVE IT TO THE LEFT AND RIGHT (DIAGONAL TO IT).

* CONTINUE IN THIS FASHION INDEFINITELY.

1

CONSTRUCTING THE TRIANGLE

1 ROW 0 1 1 ROW 1 1 2 1 ROW 2 1 3 3 1 ROW 3 1 4 6 4 1 R0W 4 1 5 10 10 5 1 ROW 5 1 6 15 20 15 6 1 ROW 6 1 7 21 35 35 21 7 1 ROW 7 1 8 28 56 70 56 28 8 1 ROW 8 1 9 36 84 126 126 84 36 9 1 ROW 9

2

PALINDROMES

EACH ROW OF NUMBERS PRODUCES A PALINDROME.

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

THE TRIANGULAR NUMBERS

ONE OF THE POLYGONAL NUMBERS

FOUND IN THE SECOND DIAGONAL

BEGINNING AT THE SECOND ROW.

1

THE TRIANGULAR NUMBERS 1 1 1 * {15} {1} 2 1 * * * 1 {3} 3 1 * * {10} * * * 1 4 {6} 4 1 * * * * * * * 1 5 10 {10} 5 1 * * * * * * * * * 1 6 15 20 {15} 6 1 * * * * {6}

* {1} * * {3} * * *

2

THE SQUARE NUMBERS

ONE OF THE POLYGONAL NUMBERS FOUND IN THE SECOND DIAGONAL BEGINNING AT THE SECOND ROW. THIS NUMBER IS THE SUM OF THE SUCCESSIVE NUMBERS IN THE DIAGONAL.

1

THE SQUARE 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

2

THE SQUARE NUMBERS

1

1 1

1 2 1

1 (3) 3 1 * * *

1 4 (6) 4 1 * * *

1 5 10 10 5 1 * * *

1 615 20 15 6 1

3

THE SQUARE 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 * * * *

4

THE SQUARE 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 * * * * *

5

THE HOCKEY STICK PATTERN

IF A DIAGONAL OF ANY LENTH IS SELECTED AND ENDS ON ANY NUMBER WITHIN THE TRIANGLE, THEN THE SUM OF THE NUMBERS IS EQUAL TO A NUMBER ON AN ADJECENT DIAGONAL BELOW IT.

1

THE HOCKEY STICK PATTERN

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 1 9 36 84 126 126 [84] 36 9 1

2

THE SUM OF THE ROWS

THE SUM OF THE NUMBERS IN ANY ROW IS EQUAL TO 2 TO THE “Nth” POWER ( “N” IS THE NUMBER OF THE ROW).

1

THE SUM OF THE ROWS

2 TO THE 0TH POWER=1 1

2 TO THE 1ST POWER=2 1 1

2 TO THE 2ND POWER=4 1 2 1

2 TO THE 3RD POWER=8 1 3 3 1

2 TO THE 4TH POWER=16 1 4 6 4 1

2 TO THE 5TH POWER=32 1 5 10 10 5 1

2 TO THE 6TH POWER=64 1 6 15 20 15 6 1

2

PRIME NUMBERS

IF THE 1ST ELEMENT IN A ROW IS A PRIME NUMBER, THEN ALL OF THE NUMBERS IN THAT ROW, EXCLUDING THE 1s, ARE DIVISIBLE BY THAT PRIME NUMBER.

1

PRIME NUMBERS

I 1 1 THE FIRST ELEMENTS IN ROWS 1 2 1 THREE, FIVE, AND SEVEN 1 *3 3 1 ARE PRIME NUMBERS. 1 4 6 4 1 NOTICE THAT THE OTHER 1 *5 10 10 5 1 NUMBERS ON THESE ROWS, 1 6 15 20 15 6 1 EXCEPT THE ONES, ARE 1 *7 21 35 35 21 7 1 DIVISIBLE BY THE FIRST 1 8 28 56 70 56 28 8 1 ELEMENT. 1 9 36 84126126 84 36 9 1

2

PROBABILITY/COMBINATIONS

PASCAL’S TRIANGLE CAN BE USED IN PROBABILITY COMBINATIONS. LET’S SAY THAT YOU HAVE FIVE

HATS ON A RACK, AND YOU WANT TO KNOW HOW MANY DIFFERENT WAYS YOU CAN PICK TWO OF THEM TO WEAR. IT DOESN’T MATTER TO YOU WHICH HAT IS ON TOP. IT JUST MATTERS WHICH TWO HATS YOU PICK. SO THE QUESTION IS “HOW MANY DIFFERENT WAYS CAN YOU PICK TWO OBJECTS FROM A SET OF FIVE OBJECTS….” THE ANSWER IS 10. THIS IS THE SECOND NUMBER IN THE FIFTH ROW. IT IS EXPRESSED AS 5:2, OR FIVE CHOOSE TWO.

1

PROBABILITY/COMBINATIONS

ROW O 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 1ROW 7 1 7 21 35 35 21 7 1

2

PROBABILITY/COMBINATIONS

HOW MANY COMBINATIONS OF THREE LETTERS CAN YOU MAKE FROM THE WORD FOOTBALL? USING THE TRIANGLE YOU WOULD EXPRESS THIS AS 8:3, OR EIGHT CHOOSE THREE. THE ANSWER IS 56. THIS IS THE THIRD NUMBER IN THE EIGHTH ROW.

3

PROBABILITY/COMBINATIONS

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 ROW 8---------1 8 28 (56) 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1

4

PASCAL’S TRIANGLE