1205 MrG What is the Binomial Expansion Theorem -- Sage

for n in range(10): expand((x+1)^n)

 $\newcommand{\Bold}[1]{\mathbf{#1}}1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 2 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{3} + 3 \, x^{2} + 3 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{5} + 5 \, x^{4} + 10 \, x^{3} + 10 \, x^{2} + 5 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{7} + 7 \, x^{6} + 21 \, x^{5} + 35 \, x^{4} + 35 \, x^{3} + 21 \, x^{2} + 7 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{8} + 8 \, x^{7} + 28 \, x^{6} + 56 \, x^{5} + 70 \, x^{4} + 56 \, x^{3} + 28 \, x^{2} + 8 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{9} + 9 \, x^{8} + 36 \, x^{7} + 84 \, x^{6} + 126 \, x^{5} + 126 \, x^{4} + 84 \, x^{3} + 36 \, x^{2} + 9 \, x + 1$

print "i\ti!" for i in range(7): print i,"\t",factorial(i)

#17) p851: (3x+1)^4 expand((3*x+1)^4)

 $\newcommand{\Bold}[1]{\mathbf{#1}}81 \, x^{4} + 108 \, x^{3} + 54 \, x^{2} + 12 \, x + 1$

def combo(n,r): return factorial(n)/factorial(r)/factorial(n-r) for r in range(5): combo(4,r)

 $\newcommand{\Bold}[1]{\mathbf{#1}}1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}4$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}6$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}4$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}1$

combo(4,0)*(3*x)^4*(1)^0

 $\newcommand{\Bold}[1]{\mathbf{#1}}81 \, x^{4}$

combo(4,1)*(3*x)^3*(1)^1

 $\newcommand{\Bold}[1]{\mathbf{#1}}108 \, x^{3}$

combo(4,2)*(3*x)^2*(1)^2

 $\newcommand{\Bold}[1]{\mathbf{#1}}54 \, x^{2}$

combo(4,3)*(3*x)^1*(1)^3

 $\newcommand{\Bold}[1]{\mathbf{#1}}12 \, x$

combo(4,4)*(3*x)^0*(1)^4

 $\newcommand{\Bold}[1]{\mathbf{#1}}1$

for r in range(5): combo(4,r)*(3*x)^(4-r)*(1)^r

 $\newcommand{\Bold}[1]{\mathbf{#1}}81 \, x^{4}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}108 \, x^{3}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}54 \, x^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}12 \, x$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}1$

1205 MrG What is the Binomial Expansion Theorem

2818 days ago by MATH4R2013