Number of monomials of degree at most d

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How does one show that the number of monomials in a finite field of degree at most d is

n+d choose n?? (sorry, I don't know how to write this in maths language)

In theory I know how to do this, but I am making a mistake somewhere along the line and get a different answer.
 
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How many choices do you have for each coefficient?
How many coefficients can you choose?

It may help to try some simple cases... if you have the field of two elements, how many quadratic polynomials are there? Cubic polynomials? You should notice a pattern pretty quickly
 
the paper I am reading says:

Finite field F[x_1, ..., x_n] of degree at most d
 

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