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I am familiar with the "stars and bars" argument to show that there are $$ {d+n-1\choose d} $$ monic monomials in $n$ variables with degree $d$.

By summing this over $d$, we find that there are $$ {d+n\choose d} $$ monic monomials in $n$ variables with degree at most $d$.

Is there an easy way to directly see this fact?

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    Why would there be any better way? It's _equivalent_ to stars and bars. – 2017-02-25
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    But you don't need to _sum_ – 2017-02-25
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    I mean, is there a way (using s&b) to see the second fact without first proving the first. – 2017-02-25
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    Yes -- just add a new fictitious variable to take up the slack (thus, a single application of stars and bars with $(n+1)$ variables). – 2017-02-25

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Just add fictive variable $\xi$ to your set of variables and apply your initial result. After that set $\xi$ to 1. Resulting monimials will have degrees at most $d$.

Say, with $\sum_{i=1}^{n+1}\alpha_i = d$

$$x_1^{\alpha_1}\dots x_n^{\alpha_n}\xi_1^{\alpha_{n+1}} \rightarrow x_1^{\alpha_1}\dots x_n^{\alpha_n}$$ with $\sum_{i=1}^{n}\alpha_i \le d.$