it’s a triangle. a triangle of numbers! pascal did not create it…. the chinese did. blaise...
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• 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?
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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!
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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!
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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
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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
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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
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_______
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
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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
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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
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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