I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial. I got stuck trying to read the Induction Proof.
They give an example of using the Sum notation:
$$ (x + y)^1 = {1 \choose 0} x^{1-0} y^0 + {1 \choose 1} x^{1-1} y^1 = \sum^{1}_{k=0} {n \choose k} x^{n-k}y^k.$$
If I simplify the terms in the middle part, I get: $ x^{1-0} y^0 + x^{1-1} y^1 = x + y $ , which is what I expected from $ (x+y)^1 $ .
However if I try to extend that to $ (x+y)^2 $ :
$$ (x + y)^2 = {2 \choose 0} x^{2-0} y^0 + {2 \choose 1} x^{2-1} y^1 + {2 \choose 2} x^{2-2} y^2. $$
And then simplify the terms, I get: $ x^{2-0} y^0 + x^{2-1} y^1 + x^{2-2} y^2 = x^2 + xy + y^2 $ . But there is only one $xy$ whereas I expected to get $2xy$ . . . as in $ (x + y)^2 = x^2 + 2xy + y^2 $ .
Why is there only $xy$ instead of $2xy$ , am I missing something here ?
The coefficients given by the formula are correct; for instance, ${2}\choose{1}$$=2$, so it gives you $x^2+2xy+y^2$. In general, ${n}\choose{k} $$=$ $$\frac{n!}{k!(n-k)!}$$ e.g. ${2}\choose{1}$$=$ $$\frac{2!}{1!\cdot1!} = \frac{2}{1} = 2$$