PASCAL'S TRIANGLE
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Row 0 |
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1 |
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Row 1 |
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1 |
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1 |
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Row 2 |
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1 |
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2 |
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1 |
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Row 3 |
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1 |
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3 |
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3 |
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1 |
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Row 4 |
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1 |
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4 |
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6 |
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4 |
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1 |
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Row 5 |
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1 |
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5 |
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10 |
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10 |
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5 |
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1 |
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Row 6 |
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1 |
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6 |
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15 |
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20 |
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15 |
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6 |
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1 |
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Row 7 |
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1 |
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7 |
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21 |
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35 |
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35 |
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21 |
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7 |
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1 |
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Row 8 |
1 |
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8 |
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28 |
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56 |
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70 |
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56 |
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28 |
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8 |
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1 |
Triangle Number: the number in the third diagonals of Pascal's triangle (red numbers).
Perfect squares: n2 = nC2 + n+1C2
Sum of the rows in Pascal's triangle: (1+1)n = 2n
n: Row number
BINOMIAL THEOREM
nCr
n: row number
r: position number
Example: 5C3 = 10
row number: 5
position number: 3
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Row 0 |
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0C0 |
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Row 1 |
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1C0 |
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1C1 |
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Row 2 |
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2C0 |
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2C1 |
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2C2 |
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Row 3 |
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3C0 |
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3C1 |
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3C2 |
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3C3 |
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Row 4 |
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4C0 |
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4C1 |
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4C2 |
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4C3 |
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4C4 |
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Row 5 |
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5C0 |
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5C1 |
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5C2 |
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5C3 |
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5C4 |
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5C5 |
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Row 6 |
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6C0 |
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6C1 |
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6C2 |
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6C3 |
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6C4 |
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6C5 |
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6C6 |
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Row 7 |
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7C0 |
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7C1 |
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7C2 |
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7C3 |
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7C4 |
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7C5 |
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7C6 |
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7C7 |
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Row 8 |
8C0 |
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8C1 |
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8C2 |
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8C3 |
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8C4 |
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8C5 |
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8C6 |
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8C7 |
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8C8 |
nCr = n-1Cr-1 + n-1Cr
(a+b)n = nC0.an + nC1. an-1. b + nC2. an-2. b2 +……..+nCr. an-r.br +……….+ nCn.bn