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PASCAL’S TRIANGLE Unit 1, Day 10

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Page 1: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

PASCAL’S TRIANGLEUnit 1, Day 10

Page 2: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

Pascal’s Wager

• “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose everything by not believing.”

Page 3: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

The Triangle

• Pascal’s Triangle is an arrangement of numbers that has fascinated mathematicians for a long time. The triangle has been found to be very useful in the study of mathematics, especially in probability theory.

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

• Q: What is the pattern used to create each row?

Page 4: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

Continued…

The rows are “row n”, where n=0, 1, 2, 3… and the diagonals are “column r” where r = 0, 1, 2, 3…

0

0

0

1

1

1

0

2

1

2

2

2

0

3

1

3

2

3

3

3

Each position of Pascal’s Triangle is

r

nis a sequence of combinations!

n = 0

n = 1

n = 2

n = 3

r = 0r = 1

r = 2

r = 3

Page 5: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

Pascal’s Identity• This gives us Pascal’s Identity:

1

1

1 r

n

r

n

r

n

We can use Pascal’s Triangle to help us determine the number of possible pathways through any grid system.

Page 6: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

Example #1: Routes on a Checkerboard

A checker is placed on a game board as shown below. Determine the number of paths the checker may take to get to each allowable square on the board if it can move only diagonally forward one square at a time. Hint: Indicate in each square the cumulative number of possible routes that led to it.

Page 7: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

Example #2: Word Arrangements• Using the arrangements of letters of the word STATS,

compute the number of paths that spell STATS, if all paths must start at the top and move diagonally down.

S T T

A A A T T T T

S S S

Page 8: PASCAL’S TRIANGLE Unit 1, Day 10. Pascal’s Wager “If God does not exist, one will lose nothing by believing in Him, while if he does exist, one will lose

Homework

p 289 #1, 7, 8, 11

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