Combinations Problem: Solving k Value in Homework Equation

[temaire]

Homework Statement

http://img234.imageshack.us/img234/8519/combgf7.png​

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Homework Equations

$${t}_k_+_1=_n{C}_kx^n^-^ky^k$$

The Attempt at a Solution

The picture I have shown contains the problem and the teacher's solution. However, I was wondering how the $$k$$ value is 3. And no, I can't ask the teacher; my test is tomorrow.

 

[rock.freak667]

This is how I would do it

$$(x^2+\frac{1}{x})^{10}=[\frac{1}{x}(x^3+<br /> 1)]^{10}$$

$$=\frac{1}{x^{10}}(x^3+1)^{10}$$

and you want the coefficient of $x^11$

so if you expand you will get

$$=\frac{1}{x^{10}}(...+^{10}C_k(1)^{10-k}(x^3)^k+...)$$

You need to find k and you want the power of x to be 11

so that 3k-10=11

See it?

 

[temaire]

$$=\frac{1}{x^{10}}(...+^{10}C_k(1)^{10-k}(x^3)^k+...)$$

You need to find k and you want the power of x to be 11

so that 3k-10=11

For your expansion, isn't the 1 supposed to be where the $$x^3$$ is? Because 1 is the y value, while $$x^3$$ is the x value.

Also, k doesn't equal 3 in $$3k-10=11$$