Tutorial: Using Pascal's Triangle for Binomial Expansion

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Tutorial: Using Pascal's Triangle for Binomial Expansion

Mathematics
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Yes Binomial theorem and Pascals triangle give the same result.
\[ \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 }\]
\[ \Large{ \text{If we record only the coefficients themselves }} \\ \Large{ \text{we have what's called the Pascal's Triangle} } \] \[\color{black}{\Large { \begin{array}{rccccccccc} n=0:& & & & & 1\\ n=1:& & & & 1 & & 1\\ n=2:& & & 1 & & 2 & & 1\\ n=3:& & 1 & & 3 & & 3 & & 1\\ n=4:& 1 & & 4 & & 6 & & 4 & & 1\\ \end{array}}}\]

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\[ \Large{ \text{Example: Use Pascal's triangle to write the expansion } }\\ \Large{ \text{ for } (x+y)^6 \text{.} } \\ \Large \text{ Directions: First make an outline of 1's } \\\Large \text{ triangle, up to n = 6.}\] \[\color{black} {\Large { \begin{array}{rccccccccc} n=0:& & & & & & & 1\\ n=1:&& && & & 1 & & 1\\ n=2:&& && & 1 & & & & 1\\ n=3:&& & & 1 & & & & & & 1\\ n=4:&& & 1 & & & & & & & & 1\\ n=5:& & 1 & & & & & & & & &&1\\ n=6:& 1 & & & & & & & & &&&&1 \end{array}}}\] \[ \\ \Large{\text{To get inner terms add the two adjacent terms above.} }\] \[\color{black}{\large { \begin{array}{rccccccccc} n=0:& && & & & & 1\\ n=1:&& & & & & 1 & & 1\\ n=2:&& & & & 1 & & 2 & & 1\\ n=3:&& && 1 & & 3 & & 3 & & 1\\ n=4:&&& 1 & & 4 & & 6 & & 4 & & 1\\ n=5:& & 1 & & 5 & & 10 & & 10 & & 5 &&1\\ n=6:& 1 & & 6 & & 15 & & 20 & & 15 &&6&&1\\ \end{array}}}\]
\[\Large \text{Using the coefficients from the row n=6,} \\ \Large \text{ each term will have the form } Cx^k y^{n-k}, \\ \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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