Question

Tutorial: Using Pascal's Triangle for Binomial Expansion

Answer 1

Yes Binomial theorem and Pascals triangle give the same result.

Answer 2

\[ \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
}\]

Answer 3

\[ \Large{ \text{If we record only the coefficients themselves }}
\\ \Large{ \text{we have what's called the Pascal's Triangle} }
\] \[\color{black}{\Large { \begin{array}{rccccccccc}
n=0:& & & & & 1\\
n=1:& & & & 1 & & 1\\
n=2:& & & 1 & & 2 & & 1\\
n=3:& & 1 & & 3 & & 3 & & 1\\
n=4:& 1 & & 4 & & 6 & & 4 & & 1\\
\end{array}}}\]

Answer 4

\[
\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.}\] \[\color{black} {\Large { \begin{array}{rccccccccc}
n=0:& & & & & & & 1\\
n=1:&& && & & 1 & & 1\\
n=2:&& && & 1 & & & & 1\\
n=3:&& & & 1 & & & & & & 1\\
n=4:&& & 1 & & & & & & & & 1\\
n=5:& & 1 & & & & & & & & &&1\\
n=6:& 1 & & & & & & & & &&&&1
\end{array}}}\] \[ \\ \Large{\text{To get inner terms add the two adjacent terms above.}
}\] \[\color{black}{\large { \begin{array}{rccccccccc}
n=0:& && & & & & 1\\
n=1:&& & & & & 1 & & 1\\
n=2:&& & & & 1 & & 2 & & 1\\
n=3:&& && 1 & & 3 & & 3 & & 1\\
n=4:&&& 1 & & 4 & & 6 & & 4 & & 1\\
n=5:& & 1 & & 5 & & 10 & & 10 & & 5 &&1\\
n=6:& 1 & & 6 & & 15 & & 20 & & 15 &&6&&1\\
\end{array}}}\]

Answer 5

\[\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.}
\]\[ \Large{ (x+y)^6 = 1x^6y^0 +6x^5y^1 +15x^4y^2 + 20x^3y^3 + 15x^2y^4 \\
~~~~~~~~~~~~~~~~~+ 6x^1y^5 + 1x^0y^6}\]