pascal 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.

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

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.

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} * * *

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.

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

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