Counting problem, given a finite field and number variables Let $F_5= {0,1,2,3,4}$ the finite field with 5 elements and let $S=F_5[x_1, x_2, x_3, x_4, x_5, x_6, x_7]$ the ring of polynomials over the $F_5$ field with 7 variables.


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*1) How many monomials of degree 5 has $S$? I solve this in this way


$\binom{5+7-1}{5}=\binom{11}{5}=462$ ways



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*2) How many monomials of degree less than or equal to 10 has $S$? I thought of the sum of the degrees 1,2,3,4,5,6,7,8,9 and 10 to calculate the answey


$\binom{1+7-1}{1}+\binom{2+7-1}{2}+\binom{3+7-1}{3}+\binom{4+7-1}{4}+\binom{5+7-1}{5}+\binom{6+7-1}{6}+\binom{7+7-1}{7}+\binom{8+7-1}{8}+\binom{9+7-1}{9}+\binom{10+7-1}{10}=19447$



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*3) How many terms of degree less than or equal to 7 is $S$? A term in $S$ is the multiplication of a field element by a monomial, ie it is an element of the form $λx_1^{a_1}⋯x_7^{a_7}$ where $λ∈F_5$ and $a_i≥0$ I do this...


vector field has 5 elements then:
$5\binom{1+7-1}{1}+5\binom{2+7-1}{2}+5\binom{3+7-1}{3}+5\binom{4+7-1}{4}+5\binom{5+7-1}{5}+5\binom{6+7-1}{6}+5\binom{7+7-1}{7}=17155$



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*4) How many polynomials of degree 5 are in $S$?





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*5) How many polynomials with exactly 3 terms are in $S$?


Good morning, I wish you can support me to see if my exercises 1,2 and 3 are correct. Also if you can guide me to solve the exercises 4 and 5, which if I understand the formation of the polynomial, but do not know how to count.
 A: Preliminary Remarks: I think of $3x_1^4x_5^1$ as a monomial. The distinction that is made between term and monomial is one that I would not make. (But it is one of the definitions of monomial given by Wikipedia.)  Also, it is not clear whether one should count $3x_1^4x_5^1$ as being "the same as" $3x_5^1x_1^4$.  
1) The number of monic monomials of degree $5$ with the $x_i$ arranged in increasing order is the number of non-negative solutions of $w_1+\cdots+w_7=5$. By the usual Stars and Bars process, this is $\binom{11}{6}$ or equivalently $\binom{11}{5}$. So I agree with your answer, if we interpret monomial as a monic term. 
2) There is a nice little trick one can use here. Introduce an eighth variable $x_8$. Then the number of monic monomials of degree $\le 10$ (and including degree $0$, that is, the monic monomial $1$) in the indeterminates $x_1,\dots,x_7$, arranged in that order, is the number of such monic monomials of degree exactly $10$ in the variables $x_1,\dots,x_8$. Using the technique of 1), you can get a very simple (non-sum) expression for this. 
3) If the variables are to be in order, then one multiplies the answer for 1) by $4$, not by $5$ as you did. 
4) One can interpret this question in a large number of ways. Are we talking about homogeneous polynomials (each term has degree $5$) or about degree $5$ in the traditional sense (some terms may have degree less than $5$)? Does order of addition matter? If it is homogeneous, and order of addition does not matter, then we want to pick out a non-empty subset of the set of terms of degree $5$. The count is then easily obtained from the answer to 3).  That is probably the interpretation you are intended to make, but the problem should have been stated in a much more precise way. 
5) As the problem is stated, there are infinitely many, indeed there are already infinitely many with $1$ term. If we put a degree restriction, and homogeneity restriction, on the polynomials then we are choosing $3$ terms.  
Remark:  We answered 4), and gave a hint for 5), on the assumption of homogeneity. Whether homogeneity should be assumed can be cleared up by asking your instructor. In two variables, for example, $x^4y+x^2+1$ has degree $5$, but is not homogeneous.  I am guessing that you are expected to assume homogeneity, but it is a guess. 
Added: We show how to compute the number of polynomials in $7$ variables, of degree $5$. The trick is the one we used for 2), but the details are more complicated. If we take such a polynomial, we can make it homogeneous by for each term multiplying by the appropriate power of a new variable $x_8$ to make each term have degree $5$. 
We know the number $T_5$ terms of degree $5$ terms in $8$ variables. So the number of homogeneous polynomials of degree $5$ in $8$ variables is $2^{T_5}-1$. (We don't use the empty set).  But the homogeneous polynomials in which each term is divisible by $z_8$ do not give a degree $5$ polynomial when we set $z_8=1$. There are $2^{T_4}-1$ of these. Thus the number of polynomials of degree $5$ in the variables $x_1,\dots,x_7$ is $2^{T_5}-2^{T_4}$.  
