This document discusses Pascal's triangle, including its namesake Blaise Pascal, how to construct the triangle by adding numbers in each row, and patterns that emerge like palindromes and polygonal numbers. It also explains how Pascal's triangle can be used for probability combinations, like calculating the number of ways to pick two hats from five hats without regard to order.

2. PASCAL’S TRIANGLE
* ABOUT THE MAN
* CONSTRUCTING THE TRIANGLE
* PATTERNS IN THE TRIANGLE
* PROBABILITY AND THE TRIANGLE

3. Blaise Pascal
JUNE 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.

4. 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.

5. 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

6. 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

7. THE TRIANGULAR NUMBERS
ONE OF THE POLYGONAL NUMBERS
FOUND IN THE SECOND DIAGONAL
BEGINNING AT THE SECOND ROW.

8. 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} * * *

9. 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.

10. 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

11. 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

12. 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 1
ROW 7 1 7 21 35 35 21 7 1