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perl
 one year ago
Tutorial: Using Pascal's Triangle for Binomial Expansion
perl
 one year ago
Tutorial: Using Pascal's Triangle for Binomial Expansion

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Pawanyadav
 one year ago
Best ResponseYou've already chosen the best response.0Yes Binomial theorem and Pascals triangle give the same result.

perl
 one year ago
Best ResponseYou've already chosen the best response.6\[ \Large \text{If we expand \( (a+b)^n\) and look at the coefficients } \\ \Large \text{ of the terms, one sees a pattern emerge. } \\ \Large \text{ Here is what you get if we expand n=0,1,2,3 and 4. } \\~\\\Large { (a+b)^0 =~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 \\(a+b)^1 =~~~~~~~~~~~~~~~~~ 1a^1b^0 + 1a^0b^1 \\(a+b)^2= ~~~~~~~~~~~1a^2b^0 +\color{red}{}2a^1b^1 + \color{red}{}1a^0b^2 \\ (a+b)^3 = ~~~~\color{black}1a^3b^0 + \color{black}3a^2b^1 + \color{black}{}3a^1b^2 + \color{black}1a^0b^3 \\ (a+b)^4 = \color{black}1a^4b^0 + \color{black}4a^3b^1 + \color{black}6a^2b^2 + \color{black}4a^1b^3 + \color{black}1a^0b^4 }\]

perl
 one year ago
Best ResponseYou've already chosen the best response.6\[ \Large{ \text{If we record only the coefficients themselves }} \\ \Large{ \text{we have what's called the Pascal's Triangle} } \] \[\color{black}{\Large { \begin{array}{rccccccccc} n=0:& & & & & 1\\ n=1:& & & & 1 & & 1\\ n=2:& & & 1 & & 2 & & 1\\ n=3:& & 1 & & 3 & & 3 & & 1\\ n=4:& 1 & & 4 & & 6 & & 4 & & 1\\ \end{array}}}\]

perl
 one year ago
Best ResponseYou've already chosen the best response.6\[ \Large{ \text{Example: Use Pascal's triangle to write the expansion } }\\ \Large{ \text{ for } (x+y)^6 \text{.} } \\ \Large \text{ Directions: First make an outline of 1's } \\\Large \text{ triangle, up to n = 6.}\] \[\color{black} {\Large { \begin{array}{rccccccccc} n=0:& & & & & & & 1\\ n=1:&& && & & 1 & & 1\\ n=2:&& && & 1 & & & & 1\\ n=3:&& & & 1 & & & & & & 1\\ n=4:&& & 1 & & & & & & & & 1\\ n=5:& & 1 & & & & & & & & &&1\\ n=6:& 1 & & & & & & & & &&&&1 \end{array}}}\] \[ \\ \Large{\text{To get inner terms add the two adjacent terms above.} }\] \[\color{black}{\large { \begin{array}{rccccccccc} n=0:& && & & & & 1\\ n=1:&& & & & & 1 & & 1\\ n=2:&& & & & 1 & & 2 & & 1\\ n=3:&& && 1 & & 3 & & 3 & & 1\\ n=4:&&& 1 & & 4 & & 6 & & 4 & & 1\\ n=5:& & 1 & & 5 & & 10 & & 10 & & 5 &&1\\ n=6:& 1 & & 6 & & 15 & & 20 & & 15 &&6&&1\\ \end{array}}}\]

perl
 one year ago
Best ResponseYou've already chosen the best response.6\[\Large \text{Using the coefficients from the row n=6,} \\ \Large \text{ each term will have the form } Cx^k y^{nk}, \\ \Large \text{ where \( C \) comes from the Pascal triangle.} \]\[ \Large{ (x+y)^6 = 1x^6y^0 +6x^5y^1 +15x^4y^2 + 20x^3y^3 + 15x^2y^4 \\ ~~~~~~~~~~~~~~~~~+ 6x^1y^5 + 1x^0y^6}\]
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