It’s a triangle. A triangle of numbers! Pascal did not create it…. The Chinese did. Blaise...

• It’s a triangle.• A triangle of numbers!• Pascal did not create it…. The Chinese did.• Blaise Pascal discovered all of the unique

patterns in it.

What is Pascal’s Triangle?

Building Pascal’s Triangle1

First we start off with a triangle of ones

1 11 1

1 11 1

Then we add the left and right number together on the second row

2

Continue with this addition for each line

3 34 46

You can keep adding rows until the cows come or your hand

hurts or your run out of paper!

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

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 0 66 12 1

1 13 78 286 715 1287 1716 1716 1287 495 66 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 1782 561 144 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 4785 2343 705 235 105 15 1

1 16 120 560 1820 4368 8008 11440 12870 11220 7128 3048 940 340 120 16 1

1 17 136 680 2380 6188 12376 19448 24310 24090 18348 10176 3988 1280 460 136 17 1

1 18 153 816 3060 8568 18564 31824 43758 48400 42438 28524 14164 5268 1740 596 153 18 1

1 19 171 969 3876 11628 27132 50388 75582 92158 90838 70962 42688 19432 7008 2336 749 171 19 1

1 20 190 1140 4845 15504 38760 77520 125970 167740 182996 161800 113650 62120 26440 9344 3085 920 190 20 1

20 Row Pascal’s Triangle

Just imagine 40 rows of a 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 1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 0 66 12 1 1 13 78 286 715 1287 1716 1716 1287 495 66 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 1782 561 144 91 14 1

A Closer Look at RowsEach row has a reference number

The very top is Row 0

0123456789

1011121314

The sum of all the numbers in a row = 2Row Number

The sum of row 6 = 26 or 64What is the sum of the eighth row?The answer is 28 or 256

Let’s Look at Elements 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 1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 0 66 12 1

1 13 78 286 715 1287 1716 1716 1287 495 66 78 13 11 14 91 364 1001 2002 3003 3432 3003 1782 561 144 91 14 1

Each number or element in a row has a reference number starting with the number 1.

The first element is always element zeroAll of these 1’s are element 0The next number in each row would be element 1

Let’s look at the 6th row!

1 6 15 20 15 6 1Element 0

Element 1

Element 2

Element 3

Element 4

Element 5

Element 6

Find the Elements 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 1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 0 66 12 1

1 13 78 286 715 1287 1716 1716 1287 495 66 78 13 11 14 91 364 1001 2002 3003 3432 3003 1782 561 144 91 14 1

Let’s find the 3rd element in 6th row

We’re at the 6th rowNow let’s go to the 3rd element 0 1 2 3

_______

Find an Element Using Math 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 1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 0 66 12 1

1 13 78 286 715 1287 1716 1716 1287 495 66 78 13 11 14 91 364 1001 2002 3003 3432 3003 1782 561 144 91 14 1

Here is the 3rd element in 6th row 1 2 3

Find 6C3 (nCr) or the 6th row choose 3rd element

r!(n-r)! n!_______

3×2×1(6-3)! 6×5×4×3×2×1 _____

6(3)! 720 _____

6(3×2×1) 720 _____

36720

= 20

“!” is a factorial. Start with the number and multiply by every sequential number down to 1

5! = 5×4×3×2×1 or 12010! = 10×9×8×7×6×5×4×3×2×1 or 3,628,800

15!5!(15-5)! _______

Theory into Practice• Let’s find the 5 element in the 15th row• We are finding nCr or 15C5.• We are using our formula with n being the row and r being the element.

5C15 =

nCr =r!(n-r)!

n!_______

1307674368000120(3628800)

_______Crunch3003

Test Your Answer 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 1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 0 66 12 1

1 13 78 286 715 1287 1716 1716 1287 495 66 78 13 11 14 91 364 1001 2002 3003 3432 3003 1782 561 144 91 14 1

Add together the two number above the 5th spot.

Go to the 15th rowNow over to where the 5th element would be

3003

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

1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 0 66 12 1 1 13 78 286 715 1287 1716 1716 1287 495 66 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 1782 561 144 91 14 1

Other Interesting Facts