Find the sum of the coefficients in the expansion ##(1+x)^n##

[RChristenk]

Homework Statement

Find the sum of the coefficients in the expansion $(1+x)^n$

Relevant Equations

Binomial Theorem

$(1+x)^n=1+C_1x+C_2x^2+C_3x^3...+C_nx^n$

Let $x=1$, hence $2^n=1+C_1+C_2+C_3...+C_n$ which is equal to the sum of the coefficients.

I originally thought the sum of the coefficients would be $2^n-1$ since the very first term $1$ is just a number and has no variable. But apparently that's not the case. So what coefficient is this $1$ for?

 

[renormalize]

So what coefficient is this $1$ for?

$1$ is the coefficient $C_0$ of $x^0$.

 

[Aurelius120]

I think
$2^n-1$ is the sum of coefficients of terms containing $x$
$2^n$ is the sum of binomial coefficients so $^nC_0$ cannot be excluded.

Since $(y+x)^n=C_0y^n+C_1y^{n-1}x^1+C_2y^{n-2}x^2.......$
$C_0$ is a coefficient of a variable $y^n$ and not a constant

 

[Mark44]

I think
$2^n-1$ is the sum of coefficients of terms containing $x$
$2^n$ is the sum of binomial coefficients so $^nC_0$ cannot be excluded.

Yes.

Since $(y+x)^n=C_0y^n+C_1y^{n-1}x^1+C_2y^{n-2}x^2.......$
$C_0$ is a coefficient of a variable $y^n$ and not a constant

Of course $C_0$ is a constant, namely 1.

 

[Aurelius120]

Yes.
Of course $C_0$ is a constant, namely 1.

I meant it's not a free constant (i.e. without a variable term). What's the correct word?

 

[Mark44]

I meant it's not a free constant without a variable. What's the correct word?

I'm not aware of any such word that means "free constant without a variable."